IN  MEMORIAM 
FLORIAN  CAJORI 


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THE    ELEMENTS 


OP 


ANALYTIC    GEOMETRY 


BY 

ALBERT  L.  CANDY,  Ph.D. 

\l 

ADJUNCT  PROFESSOR  OF  MATHEMATICS  IN  THE  UNIVERSITY  OF  NEBRASKA 


LINCOLN,  NEBRASKA 

Albert  L.  Candy 

1900 


'AH^fv-O'-hJA..-' 


Copyright,  1900 
By  Albert  L.  Candy 


CAJORI 


STATE  JOURNAL  COMPANY,  PRINTERS 


PREFACE, 


Analytic  Geometry  is  a  broader  subject  than  Conic  Sections.  It  is 
far  more  important  to  the  student  that  he  should  acquire  a  familiarity 
with  the  analytic  method,  and  thoroughly  grasp  the  generality  of  its 
processes  and  the  comprehensiveness  of  its  results,  than  that  he  should 
obtain  a  detailed  knowledge  of  any  particular  set  of  curves.  Further- 
more, all  branches  of  mathematics  are  fundamentally  and  inseparably 
related.  Any  subject,  therefore,  should  be  presented  in  such  a  way  as  to 
keep  it  in  touch  with  all  that  has  preceded,  and  at  the  same  time  reach 
forward  toward  that  which  is  immediately  to  follow,  to  the  end  that  there 
may  be  no  sudden  transition  in  passing  from  one  branch  to  another. 
Algebra  and  Geometry,  Analytics  and  Calculus  are  mutually  l^elpful,  and 
should  not  be  studied  entirely  apart.  No  one  of  these  subjects  can  be 
finished  before  the  others  are  begun.    • 

The  general  plan  and  scope  of  this  book  is  due  to  a  firm  conviction  of 
the  soundness  of  these  statements.  For  this  reason  a  fuller  treatment 
than  usual  is  given  of  the  general  analytic  method  before  taking  up  the 
study  of  the  conic  sections,  and  subjects  have  been  introduced  not 
ordinarily  treated  in  text  books  on  Analytic  Geometry.  The  method  of 
the  differential  calculus  is  the  only  way  of  studying  the  slope  of  curves, 
and  furnishes  the  best  means  of  finding  the  equation  of  the  tangent  and 
the  normal.  The  graphical  method  of  illustration  and  the  derivative  are 
indispensable  in  the  discussion  of  the  Theory  of  Equations.  The  use  of 
the  "derivative  curve"  in  the  theory  of  equal  roots,  together  with  the 
fact  that  the  ordinate  of  the  derivative  curve  is  the  slope  of  the  "integral 
curve,"  naturally  suggests  a  possible  converse  relation,  and  leads  easily 
and  logically  to  the  study  of  Quadrature,  and  Maxima  and  Minima. 

It  is  believed  that  the  elementary  discussion  of  these  subjects  here 
given  will  tend  to  meet  the  needs  of  scientific  and  engineering  students, 
who  now  require  a  knowledge  of  the  graphic  method  and  the  simple 
elements  of  the  calculus  at  the  earliest  possible  moment ;  and  that  it  will 
also  be  helpful  to  the  general  student  who  pursues  the  study  of  the 
subject  no  further.     In  the  ^^  secant  method  ^^  of  finding  the  equation 


IV  PREFACE. 

of  the  tangent  the  reasoning  is  essentially  the  same  as  in  the  method 
here  used,  but  the  student  never  comprehends  its  significance;  and 
furthermore,  he  never  uses  the  method  after  he  leaves  the  study  of  the 
conic  sections. 

Occasionally  a  subject  should  be  presented  from  the  most  general 
point  of  view,  when  the  proof  required  is  not  so  difficult  as  to  be  beyond 
the  student's  comprehension.  The  disposition,  strong  in  some  students, 
to  be  satisfied  with  a  numerical  example  or  a  special  case  should  not  be 
encouraged.  It  is  not  always  desirable  to  use  the  siniplest  demonstration 
for  a  particular  proposition,  but  rather  that  one  which  will  teach  the 
reader  to  use  the  best  method.  In  order  to  put  the  student  as  far  as 
possible  on  his  own  resources,  the  number  of  demonstrations  given  in  the 
book  has  been  reduced  to  a  minimum,  consistent  with  giving  him  a 
sufficient  working  basis,  but  these  demonstrations  have  been  made  full 
and  lucid.  For  this  reason  many  theorems  which  are  usually  proved  have 
been  left  as  exercises.  By  means  of  suggestive  questions  the  student  is 
frequently  urged  to  continue  an  investigation.  The  method  of  proof  in 
analogous  theorems  is  not  always  the  same;  and  some  important  subjects 
are  presented  from  more  than  one  point  of  view.  The  prime  object, 
which  has  been  kept  constantly  in  view,  is  to  prevent  mere  routine  work 
on  the  part  of  the  student. 

In  finding  the  equations  of  loci  special  emphasis  is  given  to  the 
meaning  of  the  parameters  which  enter  the  final  equations,  and  the 
significance  of  a  variation  in  their  value;  and  a  full  discussion  and 
thorough  geometric  interpretation  of  the  result  is  rigidly  insisted  on 
from  the  beginning.  The  teacher  should  never  lose  sight  of  this  vital 
principle. 

Polar  coordinates  and  their  relations  to  rectangular  coordinates  have 
been  introduced  at  any  early  stage.  The  transformation  of  coordinates, 
■from  the  most  general  point  of  view,  is  treated  merely  as  an  application 
of  the  distance  form  of  the  equation  of  the  straight  line. 

The  conic  section  is  first  briefly  studied  geometrically.  Its  funda- 
mental property  is  proved  in  this  way,  from  which  its  general  equation  is 
shown  to  be  of  the  second  degree.  A  short  discussion  of  the  general 
equation  of  the  second  degree  is  then  given,  not  only  for  the  purpose  of 
giving  a  general  view  of  the  conic  section  as  a  whole,  but  also  of  showing 
the  correspondence  between  the  geometric  and  the  algebraic  results,  and 
at  the  same  time  pointing  out  the  superiority  of  the  analytic  method.  The 
student  is  thus  made  to  see  that  all  possible  cases  are  wrapped  up  in  a 


PBEFACE.  V 

single  equation,  and  that  all  are  unfolded  in  a  single  investigation;  and, 
furthermore,  he  gets  a  bird's  eye  view  of  the  whole  subject. 

The  two  central  conies  are  treated  simultaneously  by  using  the  double 
sign  in  the  standard  equation.  In  this  way  much  time  is  saved;  and  the 
similarities  of  the  properties  of  the  two  conies  are  presented  in  a  striking 
manner. 

As  the  book  is  intended  for  beginners,  numerous  illustrative  examples 
are  given,  and  also  a  large  number  of  exercises.  The  numerical  examples 
have  all  been  prepared  especially  for  this  book,  and  but  few  answers  are 
submitted,  as  it  is  far  better  to  check  results  in  such  cases  by  constructing 
a  figure.  The  general  theorems  included  among  the  exercises  in  the  chap- 
ters on  the  conies  have  been  taken  chiefly  from  such  English  books  as  C. 
Smith's  Conic  Sections,  Todhunter's  Conic  Sections,  C.  L.  Loney's  Co- 
ordinate Geometry,  and  Wolstenholme's  Mathematical  Problems.  These 
have  been  selected  with  great  care.  The  object  has  been  to  include  a 
sufficient  number  of  the  easier  ones  to  prevent  the  discouragement  of  the 
poorer  students,  and  at  the  same  time  to  give  a  large  number  that  would 
test  the  power  of  the  strongest.  Altogether  the  book  contains  about  1,400 
exercises. 

If  a  short  course  is  desired,  the  sections  marked  with  a  star  can  be 
omitted.  In  any  case  these  may  be  omitted,  and  used  merely  for  the  pur- 
pose of  reference  at  the  discretion  of  the  teacher. 

I  wish  to  thank  most  heartily  all  my  colleagues  in  this  university  who 
have  so  kindly  assisted  me  in  this  work.  I  am  especially  indebted  to  Prof. 
Ellery  W.  Davis,  who  has  read  the  entire  manuscript  with  great  care,  and 
given  many  valuable  criticisms  and  helpful  suggestions. 

A.  L.  C. 
The  University  of  Nebraska, 
March  12, 1900. 


CONTEE'TS 


CHAPTER  I.    Coordinates,  Distances,  and  areas. 

SECTIONS  PAGE 

1-4.  Cartesian  coordinates.    Exercises 1 

5.  Polar  coordinates.    Exercises 5 

6.  Relations  between  rectangular  and  polar  coordinates.    Exercises 7 

7-9.  Distance  between  two  points 8 

10-13.    Areas  of  polygons 12 

Examples  on  Chapter  I ; 16 

CHAPTER  II.    Loci  AND  Their  Equations. 

14-20.  Illustrative  examples  and  definitions 18 

21-22.  Plotting  loci  of  equations.    Exercises 24 

23.  Use  of  graphic  methods.    Illustrations 29 

24-26.  Intersection  of  loci.    Exercises 33 

27-29.  Symmetry  of  loci.    Exercises 35 

90-^.  Finding  the  equations  of  loci.     Conic  Sections,  Strophoid,  Rose  of  Four 

Branches.    Exercises 40 

CHAPTER  III.    The  Straight  Line. 

40-52.    Standard   forms   of   the   equations   in   rectangular  and   polar  coordinates. 

Exercises 52 

53-^.    Equations  representing  two  or  more  straight  lines.    Homogeneous  equations. 

Exercises , 72 

59-64.    Equations  of  the  straight  line  in  oblique  coordinates.    Exercises 80 

Examples  on  Chapter  III 86 

CHAPTER  IV.    Transformation  op  Coordinates. 

65-69.    Formulae  of  transformation  in  Cartesian  and  polar  coordinates.    Exercises 92 

70-71.    Transformation  of  functions  of  two  linear  expressions.    Exercises 99 

7i-77.    Parameters  of  two  loci  as  coordinates  of  points.    Exercises 104 

CHAPTER  V.    Slope,  Tangents  and  Normals. 

78-79.  Geometric  meaning  of  the  derivative  of  the  function/(aj) .112 

80.  Examples  of  limiting  values  of  ratios  of  vanishing  quantities.    Exercises 114 

81.  Illustrative  examples  of  derivatives  and  slope  of  curves.    Exercises 116 

82-84.  General  rules  for  differentiation t 119 

85.    Tangents  and  normals 121 

Examples  on  Chapter  V '. 121 


Viii  CONTENTS. 

CHAPTER  VI.    Theort  of  Equations;  Quadrature;  Maxima  and  Minima. 

SECTIONS  PAGE 

86-101.    Theory  of  Equations.    Exercises 124 

102.    Quadrature.    Exercises 143 

103-104.    Maxima  and  Minima.    Exercises 147 

CHAPTER  vn.    Conic  Sections. 

106.  Geometric  proof  of  the  fundamental  property  of  the  conic  section 155 

107.  Classification  of  the  conic  sections.    Geometric  Exercises ' 158 

108-110.    General  equation  of  the  conic  section.    Exercises 159 

111-112.    Tangents 167 

113-115.    Pole  and  polar 169 

116-117.    Asymptotes,  similar  and  conjugate  conies.    Exercises 172 

11&-122.    Standard  equations  of  the  conic  sections 179 

CHAPTER  VIII.    The  Parabola. 

123-128.    Properties  of  the  parabola,  y-  =  4ax.    Exercises 187 

Examples  on  Chapter  VIII 196 

CHAPTER  IX.    The  Circle. 

129-134.    Properties  of  the  circle.    Exercises 201 

135-137.    Systems  of  co-axial  and  orthogonal,  circles 212 

Examples  on  Chapter  IX , 216 

CHAPTER  X.    The  Ellipse  and  Hyperbola. 

x^      v^ 

138-157.    Properties  of  the  central  conies,  -;;  ±  rs  =  1.    Exercises 221 

a-*       o^ 
Examples  on  Chapter  X 254 

CHAPTER  XI.    Polar  Equation  of  the  Conic,  the  Focus  being  the  Pole. 

158-161.    Polar  equation  of  the  conic,  directrix,  asymptotes,  tangent,  etc 263 

Examples  on  Chapter  XI 271 

CHAPTER  XII.    The  General  Equation  of  the  Second  Degree. 

162-166.    Construction  of  curves  of  the  second  order  by  solving  the  equation  with 

respect  to  one  variable.    Exercises 275 

167-169.    To  transform  the  general  equation  of  the  second  degree  to  the  standard 

fonns.    Exercises 288 

170.  Equation  of  a  conic  through  given  points 298 

171,  Equation  of  a  conic  having  two  given  lines  for  its  asymptotes 299 

Examples  on  Chapter  XII 300 

APPENDIX. 
Trigonometrical  Formulae 303 


ANALYTIC  GEOMETRY. 


CHAPTER  I. 

COORDINATES,  LENGTHS  OP  LINES  AND  AREAS  OF 
POLYGONS. 

Rectilinear  Coordinates. 

1.  Let  X'Xand  Y'Ybe  two  fixed,  non-parallel  straight  lines, 
intersecting  in  the  point  0.  Let  P  be  any  point  in  the  plane  of 
these  lines.  Draw  EP  and  QF  parallel  to  Z'Z  and  Y'  Y  respect- 
ively. 

Y 


X' 


These  distances,  i?P  and  QP,  determine  the  place  of  P  within 
the  angle  XOY.  That  is,  to  every  position  of  P  there  is  one  and 
only  one  pair  of  distances,  to  every  pair  of  distances  one  and  only 
one  position  of  P.  Moreover,  the  position  of  P  can  be  found 
when  the  lengths  of  the  lines  RP  and  §Pare  given,  and  vice  versa. 

Suppose,  for  example,  that  we  are  given  BP  =  a,  QPz=b,  we 
need  only  measure  OQ  =a  and  OB  =  b.  and  draw  the  parallels 
BP  and  QP,  which  will  intersect  in  the  required  point. 

2.  The  two  lines  RP  and  QP,  or  OQ  and  OB,  which  thus  de- 
termine the  position  of  the  point  P  with  reference  to  the  lines 

9 


2  COORDINATES.  [3. 

'  'X'XkndT'i^a^e  called  the  Rectilinear  or  Cartesian  *  Co- 

; ^  •/;  ^dilia/tiiBS;o//.th^/Point  P.     QF i&  called  the  Ordinate  of  the 

'"  "   '  point  P,  an(i' is  denoted  by  the  letter  y;  EP,  or  its  equal  0§,  the 

intercept  cut  off  by  the  ordinate,  is  called  the  Abscissa,  and  is 

denoted  by  the  letter  x. 

The  fixed  lines  X'Xand  Y'  Fare  called  the  Axes  of  Coordi- 
nates, and  their  point  of  intersection  0  is  called  the  Origin. 
When  the  angle  between  the  axes  of  coordinates  is  oblique,  as  in 
the  preceding  figure,  the  axes,  and  also  the  coordinates,  are  said 
to  be  Oblique ;  when  the  angle  between  the  axes  is  right,  the 
axes  and  the  coordinates  are  said  to  be  Rectangular. 

If,  in  the  preceding  figure,  OQ  be  a  and  OR  be  b,  then  at  P, 
x=a  and  y  =  b',  at  Q,  x  =  a  and  2/^0;  at  JU,  x  =  0  and  y  =  b', 
and  at  the  origin  0,  x  =^0  and  y  =  0. 

The  axis  X'  Xis  called  the  Axis  of  Abscissas,  or  the  x-axis  ; 
and  F'  Y  is  called  the  Axis  of  Ordinates,  or  the  y-axis. 

3.  Let  OQ  and  OQ'  he  equal  in  magnitude  to  a,  and  let  OR 
and  OR'  be  equal  in  magnitude  to  b.  Through  Q,  Q' ,  R  and  R' 
draw  lines  parallel  to  the  axes,  and  intersecting  in  F^.  F,,  F^,  F^. 


Now  at  all  of  these  four  points  re  =  a,  in  magnitude,  and  y  =  b, 
in  magnitude.     Hence  in  order  that  the  equations  x  =a  and  y  =  b 

*  This  method  of  determining  the  position  of  a  point  in  a  plane  is  due  to  the  French 
Philosopher  and  Mathematician,  Descartes.  Hence  the  name  Cartesian,  The  new 
method  was  first  published  in  1637. 

"  It  is  frequently  stated  that  Descartes  was  the  first  to  apply  algebra  to  geometry. 
This  statement  is  inaccurate,  for  Vieta  and  others  had  done  this  before  him.  Even  the 
Arabs  sometimes  used  algebra  In  connection  with  geometry.  The  new  step  that  Descartes 
did  take  was  the  introduction  into  geometry  of  an  analytical  method  based  on  the  notion 


4.]  COORDINATES.  3 

shall  determine  only  one  point,  it  is  not  sufficient  to  know  the 
lengths  of  a  and  6,  we  must  also  know  the  directions  in  which  they 
are  measured. 

In  order  to  indicate  the  directions  of  lines  we  adopt  the  rule 
that  opposite  directions  shall  be  indicated  by  opposite  signs.  It  is 
agreed^  as  in  Trigonometry,  that  distances  measured  in  the  direc- 
tions OX  (or  to  the  right)  and  OF  (or  upwards)  shall  be  consid- 
ered positive.  Hence  distances  measured  in  the  directions  OX' 
(or  to  the  left)  and  OY'  (or  downwards)  mtist  be  considered  neg- 
aiive.     Therefore 

at  Pi,  X  =  a,  y  =  b;  at  P,,  a;  =  —  a,  1/  ==  6; 

at  P3,  a;  =  —  a,  y  =  —  b;  a,t  Pi,  x  =  a,  y  =  —  b. 

Thus  the  four  points  are  easily  and  clearly  distinguished,  for 
no  two  pairs  of  values  of  x  and  y  are  the  same. 

If  all  possible  values,  positive  and  negative,  be  given  to  x  and 
to  2/,  or  in  other  words,  if  both  x  and  y  be  made  to  vary  inde- 
pendently from  —  00  to  +  00 ,  all  points  in  the  plane  will  be 
obtained.  Moreover,  to  esuihpair  of  values  of  x  and  y  there  corre- 
sponds, in  all  the  plane,  one  and  only  one  point ;  to  each  point,  one 
and  only  one  pair  of  values. 

4.  For  the  sake  of  brevity,  a  point  is  usually  represented  by 
writing  its  coordinates  within  a  parenthesis,  the  abscissa  being 
always  written  ^rsi.  Thus,  in  the  preceding  figure,  Pj,  P_,,  P3,  P4 
are  the  points  (a,  6),  ( — a,  6),  ( — a,  — 6),  (a,  — 6),  respectively. 
In  general,  the  point  whose  coordinates  are  x  and  y  is  called 
the  point  (x,y). 

As  in  Trigonometry,  it  is  convenient  to  distinguish  the  parts 
into  which  the  axes  divide  the  plane  as  first,  second,  third,  and 
fourth  quadrants. 

Because  of  simplicity  in  formulae  and  equations,  it  is  generally 
more  convenient  to  use  rectangular  axes. 

of  variables  and  constants,  which  ena.bled  him  to  represent  curves  by  algebraic  equations. 
In  the  Greek  geometry,  the  idea  of  motion  was  wanting,  but  with  Descartes  it  became  a 
very  fruitful  conception.  By  him  a  point  on  a  plane  was  determined  in  position  by  its  dis- 
tances from  two  fixed  right  lines  or  axes.  These  distances  varied  with  every  change  of 
position  in  the  point.  This  geometric  idea  of  coordinate  representation,  together  with  the 
algebraic  idea  of  two  variables  in  one  equation  having  an  indefinite  number  of  simulta- 
neous values,  furnished  a  method  for  the  study  of  loci ,  which  is  admirable  for  the  generality 
of  its  solutions.  Thus  the  entire  conic  sections  of  Apollonius  is  wrapped  up  and  contained 
in  a  single  equation  of  the  second  degree." 

[A  History  of  Mathematics  by  Florian  Cajori,  p.  185.] 


4  COORDINATES.  [4. 

Accordingly,  throughout  this  book,  except  when  the  contrary 
is  expressly  stated,  the  axes  may  be  assumed  rectangular. 

EXAMPLES. 

1.  In  what  quadrants  must  a  point  lie  if  its  coordinates  have  the  same 
sign  ?    different  signs  ? 

2.  Locate  the  points  (1,-3),  (-2,  4),  (5,0),  (-1,  -3),  (4,  2),  (0,  3). 

3.  Construct  the  triangle  whose  vertices  are  the  points  (0,  4),  (—5,  — 1) 
and  (4,-3). 

4.  Construct  the  triangle  whose  vertices  are  (4,  — 1),  (1,  2),  ( —  1,  — 3). 

5.  Construct  the  quadrilateral  whose  vertices  are  the  points  (3,  4), 
(_  1 ,  4)  ^  (—  1 ,  _  2) ,  (3,  —  2) .  "What  kind  of  a  quadrilateral  is  it  ?  Con- 
sider both  oblique  and  rectangular  axes. 

6.  Plot  the  points  (8,  0),  (5,  4),  (0,  4),  (—3,  0),  (0,  —4),  (5,  —4)  and  con- 
nect them  by  straight  lines.  What  kind  of  a  figure  do  these  six  lines  en- 
close ? 

7.  Pis  the  point  (x,  y);  Pi,  P^,  P3  are  its  symmetrical  points  with  re- 
spect to  the  X-axis,  y-axis,  and  origin,  respectively.  What  are  the  coor- 
dinates of  Pj,  P2,  P3? 

8.  The  side  of  a  square  is  2a.  What  are  the  coordinates  of  its  vertices 
when  the  diagonals  are  the  axes  ? 

9.  The  side  of  an  equilateral  triangle  is  2a.  What  are  the  coordinates 
of  its  vertices,  if  one  vertex  is  at  the  origin  and  one  side  coincides  with 
the  rc-axis  ? 

10.  Where  may  a  point  be  if  its  abscissa  is  2?    if  its  ordinate  is  — 3? 

11.  Can  a  point  move  and  yet  always  satisfy  the  condition  x  =  0?  y  =  0? 
both  the  conditions  a;  =  0  and  y  =  0? 

12.  How  must  a  point  move  so  as  to  satisfy  the  condition  x  =  —  c  ? 
2/  =  d  ?    both  these  conditions  ? 

13.  If  a  point  moves  along  either  of  the  bisectors  of  the  angles  between 
the  axes,  what  is  the  relation  between  its  coordinates  ? 

14.  Where  may  a  point  be  if  its  coordinates  satisfy  the  condition 
ic-  -j-  2/^  =  a"^  ?  What  is  the  relation  between  the  coordinates  of  a  point 
which  moves  so  that  its  distance  from  the  origin  is  always  2  ? 

15.  If  a  line  AB  is  two  units  to  the  left  of  the  y-axis,  what  are  the  co- 
ordinates of  a  point  whose  distance  from  AB  is  three  units? 

-16.  If  P  be  any  point  on  the  bisector  of  the  angle  between  the  2/-axis 
and  a  line  three  units  above  the  x-axis,  what  is  the  general  relation  between 
the  coordinates  of  P? 


5.]  COOBDINATES.  6 

Polar  Ck)ORDiNATE8. 

6.  Let  0  be  a  fixed  point  called  the  Pole,  and  OX  a  fixed  line 
called  the  Initial  Line. 

Take  any  other  point  P  in  the  plane  and  draw  OP.  The  posi- 
tion of  the  point  P  with  reference  to  the  line  OX  is  known  when 
the  distance  OP  and  the  angle  XOP  are  given. 

The  line  OP  is  called  the  Radius  Vector  of  the  point  P,  and 
will  be  denoted  by  p  ;  the  angle  XOP,  which  the  radius  vector 
makes  with  the  initial  line,  is  called  the  Vectorial  Angle  of 
the  point  P,  and  will  be  denoted  by  0, 


Then  p  and  0  are  the  Polar  Coordinates  *  of  P ;  that  is,  P 
is  the  point  (/o,  d). 

As  in  Trigonometry,  it  is  agreed  that  the  angle  0  shall  be  posi- 
tive when  measured  from  OX  counter  clockwise;  that  p  shall  be 
positive  when  measured  in  the  direction  of  the  terminal  line  of 
the  vectorial  angle  0. 

In  determining  the  position  of  a  point  whose  polar  coordinates 
are  given  the  following  direction  will  be  useful:  Suppose  I  stand 
at  0  facing  in  the  direction  OX.  To  get  to  the  point  (/o,  ^),  I 
turn  through  the  angle  0  to  the  left  or  right  according  as  0  is  pod- 
tive  or  negative,  then,  keeping  my  new  facing,  I  go  a  distance  p 
forward  or  backward  according  as  p  is  positive  or  negative^f 

*  Whenever  the  position  of  a  point  in  a  plane  is  determined  by  any  two  magnitudes  what- 
ever, these  two  magnitudes  are  the  coordinates  of  the  point.  Thus  there  may  be  an  in 
definite  number  of  systems  of  coordinates.  For  an  explanation  of  other  systems  which 
are  in  common  use  see  Chap.  I  of  Elements  of  Analytical  Geometry  by  Briot  and  Bouquet, 
translated  by  J.  H.  Boyd.  In  this  book  we  shall  mainly  use  the  two  systems  already  ex- 
plained, but  a  more  general  discussion  of  the  subject  is  given  in  §§  72-76. 

t  This  method  of  locating  points  by  means  of  coordinates  is  not  altogether  new  to  the 
student,  neither  is  it  confined  to  mathematics.  For  example,  when  we  locate  places  on  the 
surface  of  the  earth  by  means  of  their  latitude  and  longitude,  we  make  use  of  a  system  of 
rectangular  coordinates  in  which  the  axes  are  the  equator  and  some  chosen  meridian. 
When  we  say  the  city  B  is  forty  miles  north-east  of  the  city  A,  we  locate  B  with  reference 
to  A  by  means  of  a  system  of  polar  coordinates  in  which  the  initial  line  is  the  meridian 
through  A,  and  A  is  the  pole.  Let  the  student  suggest  other  familiar  examples,  if  possible. 
How  are  places  located  in  cities  ?   in  Washington,  D.  C.  ? 


6  COORDINATES.  [6. 

EXAMPLES. 
Plot  on  one  diagram  the  following  points : 

1.  (4, 30°),     (—3,  135°),     (3,  120°),     (—4,  —30°). 

2.  (5,45°),    (—4,120°),    (3,-150°),    (—6,-240°). 

3.  (a,M,    (-a,  ^TT),     (a,-|:r),     (2a,-§7r),     (_^a,-f7r),    (a,  0), 

(2a,  rr), 

4.  (5,  tan-i  5),    (—  2,  tan-^  2),    (3,  —  tan-^  3),    (—  4,  tan-^  —  1). 

5.  (a,  tan-i  2),  (a,  —  tan-^  3),  (—  a,  tan-^  i),  (  —  a,  —  tan-i  f ), 
[a,ten-^(-4)]. 

6..  Plot  the  points  (—6,  30°),  (2,  150°),  (2,  —90°)  and  connect  them  by- 
straight  lines.    What  kind  of  a  figure  do  these  lines  enclose? 

7.  Plot  the  points  (a,  60°),  (b,  150°),  (a,  240°),  (6,  —30°)  and  join  them 
by  straight  lines.    What  kind  of  a  figure  do  these  lines  enclose? 

8.  Find  the  polar  coordinates  of  the  vertices  of  a  square  whose  angular 
points  in  rectangular  coordinates  are  (3,  1),  ( —  1,  —  1),  ( —  1,  3),  (3,  — 3). 

9.  The  side  of  an  equilateral  triangle  is  2a.  If  one  vertex  is  at  the 
pole  and  one  side  coincides  with  the  initial  line,  what  are  the  polar  coordi- 
nates of  its  vertices  ?    of  the  middle  points  of  the  sides  ? 

10.  Change  "equilateral  triangle  "  to  "  square  "  in  Ex.  5. 

11.  Change  "  equilateral  triangle  "  to  "  regular  hexagon  "  in  Ex.  5. 

12.  How  must  i>  and  ^  vary  in  order  to  obtain  all  points  in  the  plane? 
(See  §  3.) 

13.  Show  that  to  each  pair  of  values  of  /»  and  ^  there  corresponds  in  all 
the  plane  one  and  only  one  point. 

14.  Show  by  plotting  the  four  points,  (3,  60°),  (—  3,  240°),  (3,  —  300°), 
( —  3,  —  120°),  that  the  converse  of  Ex.  9  is  not  true. 

15.  Show  that  in  general  the  same  point  is  given  by  each  of  the  four 
pairs  of  polar  coordinates, 

16.  Show  that  for  all  integral  values  of  n  the  same  point  (p,  0)  is  also 
given  by 

(P,  ^  ±  2n7r)    and    [— /;,  n  ±z  i2n-\-  l)7r]. 

17.  Where  does  the  point  (p,  ^)  lie  if  ^  =  0 ?    if  ^  =  tt  ?    if  p  =  2? 

18.  How  can  the  point  (p,  ^)  move  it  d  =  a?  if  p  =  a ?  where  a  and 
a  are  constants. 

19.  What  condition  must  p  and  ^  satisfy  if  the  point  (p,  0)  moves  along 
a  line  perpendicular  to  the  initial  line  ?    parallel  to  the  initial  line  ? 

20.  What  is  the  position  of  the  point  (p,  0)  if  p  =  a  cos  ^?    p  =  a  sin  (9? 


6.] 


COORDINATES. 


Kelations   Between   Rectangular  and   Polar  Coordinates. 

6.  Let  P  be  any  point  whose  rectangular  coordinates  are  x 
and  ?/,  and  whose  polar  coordinates,  referred  to  0  as  pole  and  OX 
a«  initial  line,  are  />  and  0. 


Y'  P 

Draw  FQ  perpendicular  to  OX. 

Then,  according  to  the  preceding  definitions, 

OQ=x,     QP=y,     OF  =  p,      lXOP  =  0. 

From  the  right  triangle  PQO  we  have 

Oq  =  OP  cos  XOP    and     QP  =  OP  sin  XOP. 


X  =p  cos  0. 
y  =P  sin  0. 

•r' -\- if  =  p\ 


(1) 


These  equations  (1)  express  the   rectangular   coordinates  in 
terms  of  the  polar  coordinates. 

'  From  equations  (1)  we  find  the  corresponding  equations  ex- 
pressing the  polar  coordinates  in  terms  of  the  rectangular  coor- 
dinates to  be 


y 


=  tan" 


sm  0  = 


TV  +  r' 


cos  0  = 


rx^  +  f 


(2) 


By  means  of  equations  (1)  and  (2)  formulae  and  equations  in 
either  system  of  coordinates  can  be  changed  into  the  other  system 
of  coordinates. 


DISTANCES. 


[7. 


EXAMPLES. 
1.    Change  the  equation  p^  =  a^  cos  2^  to  rectangular  coordinates. 
Multiplying  the  equation  by  p^,  and  putting  cos  20  =  cos^^  --  sin^^  gives 

p4  =  a2(p2  cos2(9  —  p2  sin2^) . 
Whence  by  substituting  equations  (1)  we  have 

Change  to  polar  coordinates  the  equations 


2.    x'  +  y'  =  2rx. 

Ans.    />  =  2r  cos  0. 

3.    x'  —  y'  =  a\ 

Ans.    p2_ci2sec2^. 

4.    (2x2  _^  22/2  —  axy  =  a\x^  -j-  y^) . 

Ans.    p«  =  a^  cos  i^. 

Transform  to  rectangular  coordinates 

5.    p2  sin  26^  =  2a2. 

Ans.    xy  =  d\ 

6.    p^  cos  id  =  a«. 

Ans.    2/2  4-  iax  =  4a2. 

Distance  Between  Two  Points. 
7.     To  find  the  distance  between  two  points  whose  rectilinear  coordi- 
nates are  given. 

Let  Pi(a?i,  i/i)  and  P2(^2?  2/2)  ^^  *^^  given  points,  and  let  the  axes 
be  inclined  at  an  angle  (o. 

Draw  Pi§i  and  F2Q2  parallel  to  OF,  to  meet  OX  in  Q^  and  Q^, 
Draw  F2B  parallel  to  OX  to  meet  PiQi  in  E, 


Then     OQ,  =  x„     OQ^^x^,     Q,P,  =  yu     Q2P2  =  y2' 
/.    P2B=Q2Qr=OQ,^OQ2  =  x^-X2, 
and  i^P,  =  Q,P,  -  q,R  =  q,P,  -  §,P,  ==  ^/i  -  2/2- 

Also  z  P,PP2=  Z  PiQiO  =  Tz  —  w, 


7.]  DISTANCES.  9 

From  the  triangle  PiRPz  we  have,  by  the  law  of  cosines  in 
Trigonometry, 

P,P{  =  P,R'  +  RP{  —2P,R'  RP,  cos  (rr  —  w). 
Whence  by  substitution,  since  cos  (r  —  w)  =  —  cos  w, 
P,P,  =[{x,-x,y  +  (y,  _y,)2  -|-2(:r.  -x.^  (y,  -y,)  C08a>]«.    (1) 
If,  as  is  usually  the  case,  the  axes  are  rectangular, 

(o  =  90°  and  cos  oj  =0. 
Hence  for  the  distance  between  two  points  whose  rectangular 
coordinates  are  given,  we  have  the  very  useful  formula 

P,P,  =  V(ix,-x,y-^(y,-y,y.^  (2) 

If  the  plus  sign  before  the  radicals  in  (1)  and  (2)  gives  Pa^o 
the  minus  sign  will  give  PxPi- 

It  will  aid  the  memory  to  observe  that  the  meaning  of  (2)  is 
expressed  by  writing 

{Distance)'^  =  (Easting)-  -\-  {NorthingY. 

Cor.  If  Pj  coincides  with  the  origin  X2  =  y-2  =  0,  and  then 
equations  (1)  and  (2)  give  for  the  distance  of  a  point  Pi(a?i,  y{) 
from  the  origin 

OP^  =  Vxx^  +  yi^  +  2  0^1 2/1  cos  w,     for  oblique  axes,  (3  ) 

OPi  =  Vxx  +  yi,     for  rectangular  axes.  (4) 

EXAMPLES. 

1.  Find  the  distance  between  (—  5,  3)  and  (7,  —  2). 

2.  Show  that  if  the  axes  are  inclined  at  an  angle  of  60°,  the  distance 
between  the  points  (—3,  3)  and  (4,  —2)  is  y  39. 

3.  Find  the  distance  from  the  origin  to  the  point  (—2,  4)  when  the  axes 
are  inclined  at  angle  of  120°. 

4.  *  Find  the  lengths  of  the  sides  of  the  triangle  whose  vertices  are  (4, 1), 
(-2,  4),  and  (1,-2). 

5.  Show  that  the  four  points  (2,  4),  (1,  7),  (—2,  4),  and  (—1,  1)  are  the 
angular  points  of  a  parallelogram. 

r6/    If  the  point  (a;,  y)  is  5  units  distant  from  the  point  (3,  4),  then  will 

a^'-h/  — ^— %=Q- 

♦The  student  should  convince  himself  of  the  generality  of  equations  (1)  and  (2)  by- 
constructing  other  special  cases  in  which  the  given  points  lie  in  different  quadrants. 
He  will  thus  have  one  illustration  of  a  general  principle  whose  truth  he  will  gradually  see 
as  he  proceeds  with  the  study  of  the  subject;  viz.  that  formulae  and  equations  deduced  by 
considering  points  lying  in  the  first  quadrant,  where  both  coordinates  are  positive,  must, 
from  the  nature  of  the  analytic  method,  hold  true  when  the  points  are  situated  in  any 
quadrant. 


10 


DISTANCES. 


[8. 


8.     To  express  the  distance  between  two  points  in  terms  oj  their  polar 
eoordiTiates. 


»  O  >^Pi  Pa 

Let  Pi(/Oj,  ^i)  and  P2(/^2j  ^2)  t>6  the  two  given  points. 
Then      OPx  =  Pu      OP,  =  p,,      Z.XOP,  =  0„      iXOP, 

and  I  P,OP,  =  0,  —  0,. 

From  the  triangle  Pj  OP2,  as  in  §  7,  we  have 

P,F/  =  0P{  +  OPi  —  2 OP,'  OP,  cos P, OP,. 


(1) 


.-.     P,P,  =  Vp,'  -f  />/  _ 2^^^,  cos  (^,  —  0,). 

Ex.  1.    Derive  equation  (2),  §  7,  from  equation  (1),  §  8. 
Expanding  the  last  term  and  squaring  (1),  §  8,  gives 

P1P2'  =  /^i'  +  P2'  —  2(/i,  cos  ^i)(P2  cos  ^2)  —  2(/>i  sin  6>,)0'2  sin  e.,). 
Substituting  the  values  given  in  equations  (1),  §  6,  we  have 
P,P^  =  x{'  +  2/1^  +  x./  +  y./  -  2x,x.  -  22/12/2. 

9.-.    P1P2  =  1/ (iC2  -  a:, )^  +  (^7-1/7)'^. 
Show  that  the  distance  between  the  points  (4,  90°)  and  (—  3,  30°) 

Ex.  3.     Find  the  distance  between  (2a,  180°)  and  (—a,  45°). 

9.      To  find  the  coordinates  of  the  point  which  divides  the  line  joining 
tivo  given  points  in  a  given  ratio  (m^-.m,). 

Let  Pi(^i,  yi)  and  P-iioc,,  2/2)  be  the  two  given  points,  and  let 
P{x,  y)  be  the  required  point. 

Draw  Pi§,,  PQ,  P2Q2  parallel  to  the  ^/-axis,  and  PE,  P^P,  par- 
allel to  the  a?- axis. 

Then  PjP,  ==  x  —  Xi,     PR  ^^x,  —  a?, 

R,P  =  y  —  y,,     EP.,  =  y2  —  y, 


From  the  similar  triangles  PiPR^  and  PP^R,  we  have 
P,P       P,R,        R,P      m, 


PP2"~    PR 


That  is, 


m, 


PP2~ 

2/  — yi 


m<, 


and 


^i(^2  —  a^)  =  m2(ic  —  xC)j 


Solving  (1)  and  (2)  for  x  and  i/,  respectively,  we  obtain 


m^x^  +  maaji 


2/  = 


m,2/2  +  may, 


(1) 
(2) 

(3) 


m^-rm^  wii  +  m2 

When  the  division  is  internal,  as  we  have  so  far  assumed,  the 
ratio  of  the  segments  is  negative;  that  is, 

PiP_       m, 

P,P~      m; 

But  when  the  division  is  external,  this  ratio  is  positive.     Hence 

for  external  division  we  have,  by  simply  changing  the  sign  of  m.^ 

in  equations  (3), 

m^x.^  —  m^y  m,2/2  —  m.;^, 


y  = 


(4) 


If  P  be  the  middle  point  of  PiP^,  ^j  =^m.^j  and  therefore  the 
coordinates  of  the  middle  of  a  line  joining  two  given  points  are 

X  =  ^{X,^X^),  yz=^(y,-}-y,).  (5) 

These  formulae,  (3),  (4),  (5),  are  independent  of  the  angle  be- 
tween the  axes,  and  therefore  they  hold  for  rectangular  as  well  as 
for  oblique  axes. 


12 


AREAS. 


[10. 


Areas  of  Polygons. 

10.   ,To  find  the  area  of  a  triangle  in  terms  of  the  coordinates  of  its 
vertices,  the  axes'  being  inclined  at  an  angle  w. 

Case  I.    When  one  vertex  is  at  the  origin. 

Y/  iY  , 


Q2  Qi  /  Px 

Let  Pi(a?i,  ?/i),  P2(x2y  y^)  be  the  other  two  vertices.     Draw 
P\Q\,  P2Q2  parallel  to  the  ?/-axis,  and  Q^R  perpendicular  to  P^Q^^, 

Then       OQ,  =  x,,  OQ,  =  x^,  Q,P,  =  y„  q,P,  =  y,, 
PQi=  Q2Q1  sin  aj  =  (xi  —  X2)  sin  <o, 
and      A  OP,P,  =  A  OQ^P.  +  trap.  Q^Q.P.P^^  —  A  OQ^Pi, 

-=  iO§,  •  Q,P,  sin  io  -f-  i§3§j(  q,P^  4-  Q^P^)  sin  u> 

—  hOQr  Q^PxSinio 
=  i[^2^2  +  (^1  —  X2)(yi  +  ?/2)  —  x,y,']  sin  a> 
=  2(^12/2  —  ^2^/1)  sinw.  (1) 

In  the  notation  of  determinants  this  may  be  written 


A  0P,P2  =  i\^''  ^Msino;. 
"  I  ^2  7  2/2 1 


(2) 


*  The  area  of  the  crossed  trapezoid  ABCD,  in  which  the  non-parallel  sides  intersect,  is 
the  difference  of  the  areas  of  the  two  triangles  formed  by  drawing  the  diagonal  AC  That 
is,  ABCD  =  ABC— ADC -^  ABE— CDE. 

This  is  expressed  analytically  by  saying  that  the  area  is  the  algebraic  sum  of  the  tri- 
angles. The  base  CD  is  then  regarded  as  having  changed  its  direction  (and  hence  its 
sign)  with  reference  to  AB;  for  in  going  along  the  sides 
consecutively  in  the  order  ABCD,  the  base  CD  is  trav- 
ersed in  the  same  direction  as  AB,  which  is  not  the 
case  in  the  ordinary  trapezoid. 

This  figure  furnishes  a  good  illustration  of  the  prin- 
ciple that  areas  gone  around  so  as  to  be  on  the  left  differ 
in  sign  from  those  gone  around  so  as  to  be  on  the 
right.    (See  §  11.) 


10.] 


AREAS. 


13 


Case  II.    When  the  origin  is  not  a  vertex  of  the  given  triangle. 
y/  ^  \Y  Pa 


Let  Pj(ic,,  iji),  P^ixo,  iji),  ACa^a,  2/3)  be  the  vertices  of  the  given 
triangle. 

Draw  the  lines  OP,,  OP.,  OP,. 
Then  by  Case  I  we  have 


A  OPiP,  =  i(a?,i/,  —  X2yi)  sin  f/>  =  ^ 


sm  to. 


A  OP,P,  =  U^,y,  —  x,y,)  sin  .>  =  J  |  ^^'  ^M  sin  w. 


Xsj  y^ 


xi,y, 


sm  a>. 


A  OP3P,  =:  K^^yi  —  Xiys)  sin  o>  =rr  ^ 

•.    A  P1P2P2  =  ilCxiy.  —  X2yi)  +  (x^y,  —  x^y^,) 

+  fe//i— «i?/:0]  sin  w        (3) 


^2  >  ^2 


+ 


•^2,    2/2^ 


'3,   2/3  I  1     gi 


sm  w 


^n2/u  1 

^2?   2/2  >   1 
^3?    2/35    I 


sin  o). 


(4) 


When  the  axes  are  rectangular  sin  </>  =  !,  and  equations  (1), 
(2),  (3),  (4),  respectively,  reduce  to 


«^2>    ^2 


(5) 


A  P1P2A  =  i(^i^2  —  ^22/1  H-  ^22/3  —  Xzy^  +  a^3/yi  —  ^12/3)      (^>) 

(7) 


I  a?3,  y%i  1 


^^  i  r^i       ^2  J  2/1       2/2 1 
I  ^2      ^3  J  2^2       2/3 ' 


14  AREAS.  [11. 

11.  When  the  origin  is  within  the  given  triangle,  the 
given  triangle  includes  the  three  triangles  OP^Py,  OP2P3,  OP^P^ 
(§  10) ;  hence  the  expressions  ^{x-^y2  —  ^2^1) >  1(^2^3  —  ^3^/2) ?  ^^^ 
^{x^yi  —  ^12/3)  must  have  the  same  sign.  When  the  origin  is  out- 
side, the  given  triangle  does  not  include  all  of  these  triangles, 
and  therefore  the  above  expressions  can  not  have  the  same  sign. 

Suppose  a  person  to  start  from  0  and  walk  consecutively  around 
the  triangles  OP1P2,  OP^P^j  OP^P^  in  the  direction  indicated  by 
this  order  of  vertices.  This  imaginary  person  would  thus  walk 
along  each  side  of  the  given  triangle  once  in  the  same  direction 
around  the  figure,  as  indicated  by  P^P^P^,  and  along  each  of  the 
lines  OPi,  OP2,  OP^  twice  in  opposite  directions.  When  the  origin 
is  inside  the  given  triangle,  he  would  walk  around  each  of  these 
triangles  in  such  a  manner  that  he  would  have  its  area  always  on 
his  left  hand.  When  the  origin  is  outside,  he  would  go  around 
those  triangles  which  include  no  part  of  the  given  triangle,  in  such 
a  manner  that  he  would  have  their  area  always  on  his  right  hand. 

Thus  direction  around  a  triangle  may  be  taken  to  indicate  the 
sign  of  its  area.   (See  note  under  §  10.) 

The  expressions  for  area  in  §  10  will  be  found  to  be  positive,  if 
the  vertices  are  numbered  so  that  in  passing  around  in  the  direc- 
tion thus  indicated  the  area  is  always  on  the  left. 

Let  the  student  show  by  trial  that 

(^i2/2  —  ^22/1)  is  ±  according  as  angle  P1OP2  is  ±  ; 
angle  P^  OP^  is  ±  according  as  the  cycle  0PiP2  is  ± . 

12.  To  express  the  area  of  a  triangle  in  terms  of  the  polar  coordi- 
nates of  its  vertices. 

Let  Pi(/Oi,  6*,),  P2(P2,  ^2),  Pz{Pzi  ^3)  he  the  three  vertices. 
Then x^  =  pi  cos  <?i ,     X2=^ p^  cos  0^^     Xs  =  p^  cos  0^^ 

i/i  =  />isin^i,    2/2  =  />2sin^2,    y^^p^sind^.     [(1),§6.] 
Substituting  these  values  in  (5)  and  (6)  of  §  10  gives 
A  OPiP2  =  ipip2  (sin  6^2  cos  <?,  — cos  02  sin  ^1) 

=  ip,P2sin  {0^  —  e,).  (1) 

A  P1P2P3  =  i  \.Pip2  sin  (^2  —  ^1)  +  P2pz  sin  (^3  —  6,) 

+  ;03/>iSin(^,-<?3)].         (2) 
From  (1)  it  follows  that  the  three  terms  of  (2)  represent,  re- 
spectively, the  areas  of  the  triangles  OP^Pi,  OPiPz,  and  OP^Pi' 


13.]  ABEAS.  15 

The  signs  of  these  terms  are  the  signs  of  the  angle  differences 
(since  p  can  always  be  made  positive),  and  we  therefore  have  an 
independent  proof  of  the  statements  in  §  11. 

Let  the  student  prove  (1)  and  (2)  directly  from  a  figure. 

1 3.     To  find  the  area  of  any  polygon  when  the  rectangular  coordi- 
nates of  its  vertices  are  known. 

Let  P,(,r,,  I/,),  P-iix.^,  yo),  ^(^3,  ys),  Pii-r^,  yd  •  -  -  -?«(««,  ?/„) 
be  the  n  vertices  of  the  given  polygon. 
Then,  we  have,  from  (5)  §  10, 


A  OP,P,  =  J  ^■'^'  ,    A  0P^3  =  i^^'^M, 

A  OPaP^^^h^^'L    A  OP,P,  =  ^l^*'2/4| 

^\x,,yA 
,\     Area  P1P2  .  .  .  P„  =  |  j|^»'  2/iU  j«2,  «/2J_^k3,  y^l 

\\'^iy    y*\  _L.  I'^nj    yn\    \  /■i\ 

since  the  area  of  the  polygon  is  the  algebraic  sum  of  the  areas  of 
these  triangles. 

This  formula  is  easy  to  remember,  but  by  expanding  the  deter- 
minants and  collecting  the  positive  and  negative  terms  it  may  be 
written, 

Area  PiPj  .  .  .  Pn  =  4  [('«i2/2  +  ^2^3  +  ^3^/*  +  •  •  •  x^yO 

—  (^1^2  +  2/2^3  +  2/3^4+  .  •  .  ynXi)'],  (2) 
which  gives  the  following  simple  rule  for  finding  the  area  of  a 
polygon  when  the  rectangular  coordinates  of  its  vertices  are 
known : 

(1)  Number  the  vertices  consecutively,  keeping  the  area  on  the  left. 

(2)  Multiply  each  abscissa  by  the  next  ordinate. 

(3)  Multiply  each  ordinate  by  the  next  abscissa. 

(4)  From  the  sum  of  the  first  set  of  products  subtra^  the  sum  of  the 
second  set  and  take  half  of  the  result. 

If  the  axes  are  oblique,  the  second  members  of  ( 1 )  and  (2 ) 
must  be  multiplied  by  the  sine  of  the  angle  between  the  axes. 
The  law  of  the  sign  of  the  area  is  the  game  as  for  the  triangle. 


16  EXAMPLES   ON   CHAPTER   I.  [13. 

Examples  on  Chapter  I. 

Find  the  area  of  the  polygons  the  coordinates  of  whose  vertices  taken  in 
order  are,  respectively, 

1.  (1,3),    (-2,-4),    and    (3,-1). 

2.  (2,5),     (-6,-2;,    and    (—1,5),    when    w=60°. 

3.  (4,15°),    (—5,45°),    and    (6,75°). 

4.  (3,-30°),    (—5,150°),    and    (4,210°). 

5.  (2,15°),    (6,75°),    and    (5,135°). 

6.  (— a,  ^-),    (a,  i^TT),    and    (_2a,  — §7r). 

7.  (a,  6 -{-<')>     (^>  ^  —  c),    and    ( — a,c)* 

8.  {afC-\-a),    (ciyC),    and    (—afC  —  a). 

9.  (2,3),    (-1,4),    (-5,-2),    and    (3,-2). 

10.  (4,1),  (1,5),  (-2,6),  (-5,3),  (-1,-1),  (-3,-4),  (1,-2), 
and(3,  —  4). 

11.  What  are  the  rectangular  coordinates  of    (4,  30°),  (—  2,  135°), 

(-3,§7r)? 

12.  What  are  the  polar  coordinates  of  (3,  —  4),     (—  5,  12),     (1,3)? 

13.  Find  the  coordinates  of  the  points  which  trisect  the  line  joining  the 
points  (—2,  —  1)  and  (3,  2). 

14.  Find  the  coordinates  of  the  point  which  divides  the  line  joining 
(3,  —  2)  and  ( —  5,  4)  internally  in  the  ratio  3  :  4. 

15.  Find  the  coordinates  of  the  point  which  divides  the  line  joining 
(5,  3)  and  ( —  1,  4)  externally  in  the  ratio  3:2. 

16.  Find  the  length  of  the  sides  and  medians  of  the  triangle  (2,  6), 
(7,  —  6),  (—  5,  —  1).    What  kind  of  a  triangle  is  it  ? 

17.  Find  the  length  of  the  sides  and  the  area  of  the  triangle  (3,  4), 
(—1,0),  (2,  —  3).    What  kind  of  a  triangle  is  it  ? 

18.  Find  the  sides  and  area  of  the  quadrilateral  whose  vertices  taken  in 
order  are  (5,-1),  (—1,  2),  (—5,  0),  and  (1,-3).  What  kind  of  a  quad- 
rilateral is  it  ? 

Change  to  polar  coordinates  the  equations 

.19.    a;2-f-2/2  =  r2.  20.    y  =  xtana. 

21.  x^  =  y\2a  —  x). 
Transform  to  Cartesian  coordinates 

22.  B  =  tan-i  m.  23.    n^  =  a'  sec  2  ^. 
24.    p  =  asin2^.  25.    i>^=- a^i  sin  ^H. 


13.]  EXAMPLES    ON    CHAPTER   I.  17 

Prove  analytically  the  following  theorems : 

26.    The  diagonals  of  a  parallelogram  bisect  each  other. 

^^J  The  lines  joining  the  middle  points  of  the  adjacent  sides  of  any 
quadrilateral  form  a  parallelogram. 

28.  The  three  medians  of  a  triangle  meet  in  a  point,  which  is  one  of 
their  points  of  trisection. 

29.  The  area  of  the  triangle  formed  by  joining  the  middle  points  of  the 
sides  of  a  given  triangle  is  equal  to  one-fourth  of  the  area  of  the  given 
triangle. 

/^/  If  in  any  triangle  a  median  be  drawn  from  the  vertex  to  the  base,  the 
sum  of  the  squares  of  the  other  two  sides  is  equal  to  twice  the  square  of 
half  the  base  plus  twice  the  square  of  the  median. 

^Vr  The  sum  of  the  squares  of  the  four  sides  of  any  quadrilateral  is  equal 
to  the  sum  of  the  squares  of  the  diagonals  plus  four  times  the  square  of 
the  line  joining  the  middle  points  of  the  diagonals. 

32.  The  lines  joining  the  middle  points  of  opposite  sides  of  any  quadri- 
lateral and  the  line  joining  the  middle  points  of  its  diagonals  meet  in  a 
point  and  bisect  one  another. 

33.  P,(x„3/,),  P,(aJ2, 2/2),  -PsCaJs,  2/3),  PtCa;*,  2/0  •  •  •  P«(a;„,  2/,.)  are  any 
w  points  in  a  plane.  P1P2  is  bisected  atQ,;  Q1P3  is  divided  at  Q2  in  the 
ratio  1:2;  Q^Pi  is  divided  at  Q3  in  the  ratio  1:3;  Q^Pi  at  Q^  in  the  ratio  1 : 4, 
and  so  on  till  all  the  points  are  used.  Show  that  the  coordinates  of  tha 
final  point  so  obtained  are 

X,-\-X^-\-X^-\-X,-{-     .    ,    .     Xn         ^^^         2/1+^2  +  2/3+^4+     '    '    '    Vn 

n  n 

Show  that  the  result  is  independent  of  the  order  in  which  the  points  are 
taken. 

[This  point  is  called  the  Centre  of  Mean  Position  of  the  n  given  points.] 


CHAPTEH  II. 


LOOI  AND  THEIR  EQUATIONS. 


14.  It  has  been  shown  in  §  3  that  to  each  pair  of  values  of  x 
and  7/  there  corresponds  in  all  the  plane  one  and  only  one  point, 
and  that  to  each  point  corresponds  one  and  only  one  pair  of 
values.  Also,  if  x  and  y  vary  independently  and  unconditionally 
from  —  00  to  00  every  point  in  tho  plane  will  be  obtained. 

If)  on  the  contrary,  one  or  both  of  the  coordinates  canno*  take 
all  values,  or  if  all  values  cannot  be  independently  taken  by 
both,  the  point  cannot  move  to  all  positions  in  the  plane. 


t<0 


':>0 


*<0 


If,  for  example,  a^  >  0,  the  point  {x,  y)  must  lie  to  the  right  of 
the  ?/-axis  ;  if  ic  <  0,  the  point  must  lie  to  the  left  of  the  i/-axis  ; 
if  x  is  neither  greater  nor  ?ess  than  zero,  the  point  can  lie  neither 
to  the  right  nor  to  th6  lejt  of  the  7/-axis  ;  ^.  e.,  if  ic  =  0,  the  point 
must  lie  on  the  ?/-axis. 

Ex.    Where  must  the  point  (x,  2/)  lie  if  2/ >  0?    2/<0?    2/  =  0? 

15.  If  a?>a,  the  point  (ir,  ?/)  must  lie  to  the  right  of  the 
parallel  AB^  which  is  a  units  to  the  right  of  the  ^/-axis  ;  if  a"  <  a, 
the  point  must  lie  to  the  ^e/i5  of  AB.  Therefore,  if  ic=:a,  the 
point  will  lie  on  the  line  AB. 


Ki.l 


LOCI    AND    THEIR    EQUATIONS. 


19 


Y 

A 

*<« 

" 

,f 

O 

x<a 

1 

Y' 

B 

Ex.  1.    Where  will  the  point  {Xy  y)  lie  if  a;  >  —  3  ?    x  <  —  H  ?    x  ^  —  8  ? 

Ex.2.    Where  is  the  point  (x,  ?/)  if  ?/ :.  6  ?    y  <^  b?    y-^b?    y^~b? 
y<-b?    y=—b? 

16.     Draw  a  circle  with  centre  at  the  origin  and  radius  eqiial 
to  a. 

Y 


Then  the  point  P(jc,  y)  will  be  outside,  inside,  or  on  this  circle 
accordinj:  as 

OP>a,      OP<a,     or     OP=a, 

But  OP^  =  x'  +  y\  [(4),  §  7.] 


20 


LOCI   AND   THEIR   EQUATIONS. 


[17. 


Therefore  the  point  P(x,  y)  is  outside,  inside,  or  on  the  circle 
according  as 

^'-r//'>«^     x^-{-y-<a\     or     x''-\-y'  —  a\ 

Ex.  1.    Write  down  the  conditions  that  the  point  (x,  y)  shall  be  outside, 
inside,  or  on  the  circle  whose  centre  is  at  the  origin  and  radius  3. 

Ex.  2.    What  are  the  conditions  that  the  point  (a:,  y)  shall  be  outside, 
inside,  or  on  a  circle  with  centre  at  ( — 3,  1)  and  radius  4? 

Ex.  3.     Draw  a  circle  with  centre  at  (a,  6)  and  radius  r,  and  write  down 
the  conditions  that  the  point  (a:,  y)  shall  be  outside,  inside,  or  on  this  circle. 

1 7.     Let  the  line  A  OB  bisect  the  angle  XO  Y. 


Then  every  point  on  AB  is  equidistant  from  the  axes.  Hence 
the  point  P{x,  y)  is  above  AB,  below  AB,  or  07i  AB  according  as 

y>x,     y<x,     or     y  =  x, 

or  according  as  y  —  x^,   <,  or  =  0  ; 

i.  e.  according  as  i/  —  ;>c  is  positive,  negative,  or  zero. 

Ex.  What  are  the  conditions  that  the  point  (x,  y)  shall  be  above,  below, 
or  on  the  bisector  of  the  angle  X^OY? 

1 8.  Draw  CD  parallel  to  AB,  cutting  the  y/-axis  in  E,  three 
units  above  0. 

Then  every  point  on  CD  is  three  units  farther  from  the  x-ax's 
than  from  the  ?/-axis.     Therefore  the  poii.t  P(x,  y)  will  be  above 


19.]  LOCI   AND  THEIR   EQUATIONS. 

CD,  below  CD,  or  on  CD,  according  as 

!/>,    <,   or  =.r +  3; 
/.  e.  according  as  y  —  x  —  3  is  positive,  negative,  or  zero. 


21 


Y 

A 

P  y^ 

X            J 

-^  ^ 

£ 

X 

/ 

/ 

f 

y 

3) 

/ 

y 

/ 

y 

c/          A 

O 

Y' 

Ex.  1.  Draw  a  line  parallel  to  ABy  cutting  the  i/-axis  two  units  below  O; 
and  write  down  the  conditions  that  the  point  (x,  y)  shall  be  above,  below, 
or  on  this  line. 

Ex.  2.  What  are  the  conditions  that  the  point  (x,  t/)  shall  be  above,  be- 
low, or  on  the  line  through  ^parallel  to  the  bisector  of  the  angle  X^OV^ 

Ex.  3.    "Where  is  the  point  (x,  j/)  if  3/  +  x  -j-  4 >,    <,    or    =0? 

Ex.4.    Locate  the  point  (x,  3/)  if  3/ — 2x  —  2>,     <,    or    =0. 

Ex.  5.    Locate  the  point  (x,  y)  if  2y  -f-  3x  —  1  >,    <,    or    =  0. 

19.  Let  CD  be  the  perpendicular  bisector  of  the  line  joining 
Jl(— 1,  1)  and  J5(3,  —1). 

Then  all  points  on  CD  are  equidistant  from  A  and  B,  and  all 
other  points  are  not  equally  distant  from  A  and  B.  Hence  the 
point  P{x,  y)  will  lie  to  the  right  of,  to  the  left  of,  or  on  CD 
according  as 

AP>,   <  or  =BP, 
or  according  as 

AP'->,   <,  or  =BP'', 


i.  e.  according  as  [(2),  §  7] 

(x-{-iy-^(y  —  iy>,   <,  or  =(,r- 
whence  2x  —  y  —  2>,   <,  or  = 


-sy-^(y-^iy 
0. 


22 


LOCI    AND   THEIR   EQUATIONS. 


[20. 


Y 

^ 

P 

^' 

A^ 

^ 

^ 

h^ 

\ 

A 

O 

/ 

^\ 

4/ 

B 

c/ 

Y' 

Ex.  1.  Find  the  conditions  that  the  point  (x,  y')  shall  be  above,  below, 
or  on  the  perpendicular  bisector  of  the  line  joining  (2,  3)  and  ( —  1,  — 2). 

Ex.  2.  "What  is  the  condition  that  (x,  y)  shall  be  on  the  perpendicular 
bisector  of  the  line  joining  (a,  6)  and  c,  d)  ? 

20.  The  foregoing  examples  (§§  14-19)  illustrate  certain  gen- 
eral principles,  of  which  we  will  here  make  only  a  preliminary 
statement. 

I.  All  points  whose  coordinates  satisfy  an  equation  of  condition 
(not  an  identity)  lie  on  a  certain  line  ;  and  conversely,  if  a  point 
lies  on  a  fixed  line,  its  coordinates  must  satisfy  an  equation, 

II.  Points  whose  coordinates  satisfy  a  condition  of  inequality 
do  not  lie  on  any  fixed  line. 

If /(a?,  ?/)  be  used  to  represent  any  expression  (which  is  not  de- 
composable) containing  the  two  variables  x  and  y  and  certain  con- 
stants, these  principles  may  be  stated  more  definitely,  as  follows : 

I.  All  points  whose  coordinates  make  /(a?,  y)  =  0,  lie  on  a 
certain  line  ;  and  conversely,  the  coordinates  of  all  points  on  this 
line  make/(;r,  y)  =  0. 

II.  If  /(.Xi,  i/i)  >  0  a,ndf(x2,  y^)  <  0,  the  two  points  (;r,,  t/,) 
and  (iTa,  2/2)  li®  on  opposite  sides  of  the  line  the  coordinates  of 
whose  points  make /(a;,  y)  =0. 

Hence  every  line,  as  well  as  the  axes  of  coordinates,  is  said  to 
have  a  positive  and  a  negative  side. 


20.]  LOCI    AND    TlIEIll    EQUATIONS.  28 

Def.  The  locus  of  a  variable  point  subject  to  a  given  condition  is 
thf  place,  i.  e.  the  totality  of  positions,  where  the  point  may  lie  and  sat- 
isfy the  given  condition. 

Def.  The  line  (or  lines)  containing  all  points,  and  no  others,  whose 
coordinates  satisfy  a  given  equation  is  called  the  LocUS  Of  the  Equa- 
tion ;  conversely,  the  equation  satisfied  by  the  coordinates  of  all  points 
on  a  certain  line  (or  lines)  is  called  the  Equation  of  the  Line,  or 
the  Equation  of  the  Locus. 

Def.  That  part  of  the  plane  containing  all  points,  and  no 
others,  whose  coordinates  satisfy  a  given  inequation  is  the  Locus 
of  the  Inequation. 

Thus  the  Locus  of  a  point  in  Plane  Geometry  is  not  always  a 
line. 

In  the  examples  of  §§  14-19  only  Cartesian  coordinates  have 
been  used,  but  the  fundamental  principles  there  illustrated,  and 
also  the  above  definitions,  hold  for  all  systems  of  coordinates. 

Let  the  student  give  some  similar  illustrations  with  polar  co- 
ordinates. 

EXAMPLES. 
What  is  the  locus  of 

3.  i>  =  a  sec  ^  ?    fj  >  a  sec  H?    i>  <  a  sec  H  ? 

4.  i)=bC8cft?    i>  >  bcscf^?    f>  <  bcscfi? 

5.  4<a:2  +  2/2<9? 

6.  9<(a:  — 2)2  +  (2/  — 3)2<16? 

7.  a  sec  ^  <  /^  <  5  sec  ^  ? 

8.  i>  =  a  cos  H?    i>  >  a  cos  H?    i>  <  a  cos  ^  ? 

9.  a  cos  fi  <  i>  <b  cos  w  ? 

10.  i>  =  asinH?    /;>asin^?    pKasinft? 

11.  f>  =  a?    ()>a?    p<a? 

12.  What  is  the  locus  of  a  point  moving  so  that  the  sum  of  its  distances 
from  the  lines  x  =  0  and  a^  =  3  is  1,  2,  3,  4  ? 


24 


LOCI   AND    THEIR   EQUATIONS. 


[21. 


To  Find  the  Locus  of  a  Given  Equation. 

21.  If  the  locus  of  an  equation  is  a  straight  line,  the  locus  is 
easily  drawn  ;  it  is  only  necessary  to  locate  two  points  on  it 
(preferably  the  intersections*  with  the  axes)  and  draw  a  straight 
line  through  these  points. 

Likewise,  if  the  locus  is  a  circle,  the  complete  locus  can  be 
drawn  when  the  centre  and  radius  are  known. 

It  will  be  shown  farther  on  that  straight  lines  and  circles  can 
easily  be  recognized  by  the  forms  of  the  equations. 

In  general,  having  given  an  equation  of  condition  between  the 
coordinates  (in  any  system)  of  a  variable  point,  we  may  assign 
any  value  we  please  to  one  coordinate  and  find  a  corresponding f 
value,  or  values,  of  the  other.  To  every  such  pair  of  corre- 
sponding values  will  correspond  a  definite  point  of  the  locus. 
Since  these  pairs  of  values  may  be  as  numerous  as  we  please,  we 
can  in  this  way  locate  as  many  points  of  the  locus  as  we  please. 
A  smooth  curve  drawn  through  these  points  will  be  an  approod- 
motion  to  the  locus  of  the  given  equation.  The  degree  of  approxi- 
mation will  depend  upon  the  proximity  of  the  points  thus  lo- 
cated. This  method  of  constructing  a  locus  is  applicable  to  any 
equation  that  can  be  solved  for  one  of  the  variables,  and  is  called 
Plotting  t  an  Equation,  or  Plotting  the  Locus  of  an 
Equation.     The  steps  of  this  process  are  as  follows: 

*  Unless  both  intersections  are  near  the  origin,  when  the  line  will  be  inaccurately 
determined,  or  both  at  the  origin,  when  its  direction  will  be  quite  undetermined. 

t "  Corresponding  values  "  of  the  variables,  x  and  y  say,  involved  in  a  given  equation 
are  a  pair  of  values  of  x  and  y  which  satisfy  the  equation. 

J  The  logic  of  the  process  of 
plotting  is  that  of  induction,  and 

should  be  so  recognized  by  the  -  ^  *, 

student.    Given  the  points  A,B,  '      \  /•.       •.  ,'' 

C,D,E,F  on  a  curve;  then,  in 
the  absence  of  further  knowledge, 
we  take  as  a  probable  approxi- 
mation a  smooth  curve  drawn 
through  them  like  the  full  curve 
in  the  figure.  We  are  not  war- 
ranted in  drawing  such  a  curve  as 
the  dotted  one  through  the  points, 
because  it  is  unlikely  that,  taking 

points  at  random  on  such  an  ir-  /^,' 

regular    curve,   the    position   of 

these  points  should  fail  to  disclose  any  of  the  irregularity.  The  student  should  also  be 
warned  that  sudden  changes  of  slope  or  curvature  are  as  unlikely  as  sudden  changes  in 
the  value  of  an  ordinate. 

For  example,  the  curve  y  =  Binx  is  not 


22.] 


LOCI   AND   THEIR   EQUATIONS. 


25 


(1)  Solve  the  equation  with  respect  to  one  of  the  coordinates. 

(2)  Assign  to  the  other  coordinate  a  series  of  values  differing 
but  little  from  each  other. 

(3)  Find  each  corresponding  value,  or  values,  of  the//>/  co- 
ordinate. 

(4)  Locate  the  point  corresponding  to  each  pair  of  correspond- 
ing values  thus  found. 

(5)  Join  these  points  in  order  by  a  smooth  curve,  and  this 
curve  will  be  an  approximation  to  the  required  locus.  If  there 
be  doubt  how  to  fill  up  any  of  the  intervening  spaces,  more  points 
must  he  interpolated. 


22.     Illustrative  Examples. 
Ex.  1.    Plot  the  locus  of  the  equation  lOy  =  x^  —  3x  —  20. 
Assigning  to  x  yalues  from  —  8  to  -|- 10,  differing  by  two  units,  we  find 
the  following  pairs  of  values  of  x  and  y  to  satisfy  the  equation : 


y  = 


x  = 


y  = 


-8 

- 

-6 

—  i 

I 

-2 

6.8 

3.4 

.8 

—  1 

2 

4 

6 

8 

—  2.2 

—  1 

.6 

- 

-.2 

S 

. 

Plotting  the  corresponding  points 
Pi,  P2,  P3,  etc.,  and  drawing  a  smooth 
curve  through  them  in  the  order  of 
the  increasing  values  of  x,  we  find  the 
locus  to  be  approximately  the  curve  drawn  in  the  figure. 


■\  /v. 
Kv /\         X 


Ex.2. 


to 


00, 


Plot  the  locus  of  the  equation  y^  =  4x. 

Solving  for  y  gives  3/  =  ±  2|/ar. 

When  X  =  0, 1,  4,  9,  ...  to  00, 
3/  =  0,±2,±4,rb6  ... 
respectively. 

The  corresponding  points  of  the  locus  are 
0(0,0),  P.(l,-2),  P(l,2),  P3(4,-4), 
P4(4,4),    P5(9,— 6)    and    P,{9,6).  .  .  . 

When  X  is  negative,  y  is  imaginary. 
Therefore  no  points  of  the  locus  lie  to  the 
left  of  the  2/-axi3.  For  every  positive  value 
of  X  there  are  two  values  of  y  numerically 
equal  but  opposite  in  sign.  Hence  the  two 
corresponding  points  of  the  locus  are  equi- 
distant from  the  x-axis.  As  x  increases, 
both  values  of  y  increase  numerically. 


R^ 

X 

o 

-p-^ 


26 


LOOI    AND   THEIR   EQUATIONS. 


[22. 


Therefore  the  locus  can  not  be  such  a  curve  as  that  represented  by  the 
dotted  line,  but  must  be  approximately  that  indicated  by  the  full  line. 

Ex.  3.    Plot  the  locus  of  the  equation  25{x  —  l)^-{- 16(2/  —  3)^  =  400. 
Solving  this  equation  for  y  gives 

2/=3zbf  V  16— (x  — 1)2. 

This  form  of  the  equation  shows  that  y  is  imaginary  when  x  <  —  3,  or 
X  >  5,  since  16  —  (x  —  1  )Ms  then  negative ;  and  when  x  is  neither  less  than 
— 3  nor  greater  than  5  there  are  two  real  unequal  values  of  y,  one  found 
by  using  the  -j-  sign  before  the  radical,  the  other  found  by  using  the  — 

sign.    Herifce  the  locus  lies  between  the 
two  parallel  lines  x  =  — 3  and  a:  =  5. 

The  equation  is  satisfied  by  the  follow- 
ing pairs  of  values  of  x  and  y : 


/ 

^ 

T 

^ 

\ 

p, 

/ 

\ 

/ 

\ 

1 

\ 

p 

\ 

/ 

\ 

/ 

p. 

\ 

o 

y 

^N 

V. 

^ 

/^" 

X  = 

= 

— 

3 

—2 

—  1 

0 

y  = 

3 

6.3 

7.3 

7.8 

y  = 

3 

-.3 

-1.3 

-1.8 

x  = 

1 

2 

3 

4 

5 

y  = 

8 

7.8 

7.3 

6.3 

3 

y  = 

-2 

- 

-1.8 

—  1.3 

-.3 

3 

The  corresponding  points  are  P( — 3, 3 ) , 
Pi(— 2,  6.3),  P2(— 2,  —  .3),  etc.,  and 
the  locus  is  approximately  as  shown  in 
the  figure. 

Ex.  4.    Plot  the  locus  of  the  equation  p  =  2a  sin  B. 

Here  />  has  its  greatest  value  when 
sin  ti  has  its  greatest  value,  i.  e.  when 
d=\TT.  As  p  increases  from  0  to  ^tt, 
sin  ^  increases  from  0  to  1,  and  p  in- 
creases from  0  to  2a ;  as  /^  increases 
from  ^TT  to  -,  sin  ^  decreases  from  1 
to  0,  and  p  decreases  from  2a  to  0* 
Hence  the  locus  starts  from  the  origin 
and  returns  to  the  origin  as  B  is  made 
to  vary  from  0  to  -. 

Assigning  to  0  values  from  0  to 
180°,  differing  by  30°  we  find  the  fol- 
lowing points  are  on  the  locus : 

0(0,0),       ^(a,30°),       i5(av'3, 60°), 
^(a,  150°),    and    0(0,180°). 

The  complete  locus  is  the  curve  shown  in  the  figure. 

Ex.  a.  Show  that  the  points  A,B,  .  .  .all  lie  on  a  circle  tangent  to 
OX  at  O  and  whose  radius  is  a.  Show  also  that  every  point  on  this  circle 
satisfies  the  given  equation. 


O(2a,90°),        I>(.av/3,  120° 


22.] 


LOCI    AND   THEIB  EQUATIONS. 


27 


Ex.  6.  Show  that  the  same  circle  will  be  described  as  ^  varies  from  180° 
to  360° ;  also  as  ti  varies  from  any  value  a  to  a-\-Tr. 

We  have  in  this  example  an  illustration  of  a  characteristic  property  of 
equations  in  polar  coordinates  containing  a  periodic  function  of  (K  In 
such  equations  /)  takes  all  possible  values  as  (>  varies  through  a  limited 
range  of  values  called  the  period  of  the  function.  The  complete  locus  is 
described  at  least  once  as  0  varies  through  this  period,  and  is  repeated  as 
^  varies  through  any  other  equal  period. 

The  period  of  sin  f'  is  2:r;  hence  in  the  above  equation  />  takes  all  possi- 
ble values  from  —  2a  to  +  2a  as  ^  varies  from  0  to  27r.  The  whole  circle 
is  described  tvnce  as  ^  varies  through  this  period,  once  as  0  varies  from  0 
to  T  with  p  positive,  and  once  as  H  varies  from  -  to  2t  withp  negative.  Also 
the  whole  circle  is  described  twice  if  f)  starts  from  any  value  and  varies 
through  27r  in  either  direction. 

Ei.  5.    Plot  the  locus  of  the  equation  p  =  sin  2  B, 

Tills  equation  is  satisfied  by  the  following  pairs  of  values  of  p  and  B: 
^  =  45°,  225°,    p  =  1. 
^  =  135°,  315°,    p  =  —  \. 
^  =  30°,  60°,  210°,  240°, 

p  =  hy  3. 

i^  =  120°,  150°,  300°,  330°, 

i>  =  —  ^v  3. 

*i  =  15°,  75°,  195°,  255°, 
/>  =  L 

//  =  105°,  165°,  285°,  345°, 

P=-h. 

/^  =  0°,  90°,  180°,  270°,  360°, 

P=0. 

The  corresponding  points  are  found  by  drawing  three  circles  with  cen- 
tres at  O  and  radii  \,  ^v^3,  and  1,  and  then  drawing  radii  dividing  these 
circles  into  arcs  of  15°. 

The  locus  is  the  four-leaf  curve  shown  in  the  figure. 

As  0  varies  from  0  to  27r,  the  four  leaves  are  described  in  the- order  1,  2, 
3,  4,  and  in  the  direction  indicated  by  the  arrow  heads. 

EXAMPLES. 

Plot  the  loci  of  the  following  equations:* 

r  2a:  — 32/  — 6  =  0.    W 

1.    \  4x  — 62/  — 6  =  0.     I  2. 

i6a:  — 92/  +  27  =  0.  J 


2a: -I- 3y -1-5  =  0. 
3a:  — 22/  — 12  =  0. 
5x  +  22/  —  4  =  0. 


*  For  convenience  in  plotting  loci  in  rectangular  coordinates  the  student  should  be  sup- 
plied with  "coordinate  paper." 

t  Loci  grouped  under  the  same  number  should  be  plotted  on  the  same  diagram. 


28  LOCI    AND  THEIR    EQUATIONS.  [22. 

f2x-\-9y  +  iS  =  0.^  4.     (a;-4)(3/  +  3)  =0. 

3.    I  y  =  lx—3.  I  5.    (a;2-4)(2/  — 2)  =  0. 

7.    6x-'-\-5xy  —  6y^  =  0.  f  ar-^-f- 2/^  =  25.  ) 


|a;2  +  2/^  =  4.| 
•     ^2-2/2  =  4.  j 

r4(a:+l)  =  (2/-2)^| 

I  2/2  _  (^2  _  4)2  J 


9.    j  (x-8)'^  +  (2/-4)'^  =  25. 

11.    2/  =  a^  — 4a:2  — 4a;+16. 

12      f  2/  =  ^*  — 20^'  +  64.  I 
(2/2  =  x*  — 20x2  +  64.  i 

14.    (x2-|-2/')"'  =  aV  — 2/'-^). 


15.  y  =  Xf    x^j    3?^    x*j    a^  .  .  .  X"  .    What  points  are  common  to  these 
curves  ?    Consider  the  case  n  =  oo  . 

16.  y  =  (x  —  l),    (x—lYy    {x—lf...  Compare  with  No.  15. 

17.  2/^  =  Xf    x^,    a^f    X*.                      18.  y  =  sin  x,    cos  x. 
19.    y  =  tan  ic,    cot  x.                           20.  2/  =  sec  x,    esc  x. 
21.    ^>  =  sin  ^,  cos  /?,  sec  ^,  esc  ^.      22.  />  =  sin  3^,    sin  40, 
23.  >^  =  cos20,    cos  30,    cos  4^.        24.  />  =  tan0,    cot^. 
25.    /!>  =  sin^0,    COS  if).                     26.  p  = 


cos  0     3  —  2  cos  ^ 

27.  Are  the  points  (2,9),   (1,5),   (— 1, — 4)  on  the  same  or  opposite 
sides  of  the  locus  of  y  —  3x  =  2  ? 

28.  Are  the  points  (9,  —  10),  (5,  12),  ( —  8,  10)  on,  inside,  or  outside  the 
circle  a;2  + 2/2  =  169? 

29.  Are  the  points  (3,  60°),  (f,  — 90°)  on  the  same  or  opposite  sides  of 
the  loci  of  Ex.26? 

30.  Which  of    the  loci  represented  by  the  following  equations  pass 
through  the  origin  ? 

(1)  2x+Sy==0.     (4)    2/'  — a'a;2  =  0.  (7)    y''  =  iax. 

(2)  a;2+2/''  =  l-       (5)     ax-\-hy-\-c  =  Q,      (8)    2/'  =  4a(a:  +  a). 

(3)  y  =  3x^—x,      (6)    aa;2 -1-62/2  =  1.  (9)    (x— a)2  +  (2/— 6)2  =  a2  — 52. 

What  is  the  necessary  and  sufficient  condition  that  the  locus  of  an  equa- 
tion in  Cartesian  coordinates  shall  pass  through  the  origin  ? 

Plot  the  following  loci : 

31.  /)^  =  sin20,    cos  2^.  32.    //''  =  see2^,    esc  2^. 


23.]  LOCI    AND   THfilB   EQUATIONS.  29 

X—2       {x—\){x—2) 


33.    2/  = 


X-Z' 


{x-\){x-2)       (a:-l)(a:-3) 
"^^    ^~  (x  — 3)(x-4)'     (a;  — 2)(x-4)' 

x  +  2      (x-l)(a:-3) 

(a;.|-l)(x-2)      (a;  +  2)U-4) 
^-    2/-(a;4.3)(a;_4)'     (x-l)(x-3)' 

(a;-l)(x  — 3)(a;  — 5)      (x+ l)(a;  — 4)(a:-6) 
^~(a:  — 2)(a:  — 4)(a:  — 6)'    (ar -l)(x  +  2)(a;  — 3) ' 

(x-l)(a;-3)(a;-5>      (x- l)(a:  +  3)(x-5) 
^*    ^~       (a;-2Kx-4)      '  (a:-2)(x-4) 


The  Use  of  Graphic  Methods. 

23.  It  has  been  shown  in  §§  14-20  that  whenever  the  relation 
between  two  quantities,  whose  values  depend  upon  one  another, 
can  be  definitely  expressed  by  an  equation,  then  the  geometric  or 
graphic  representation  of  this  relation  is  given  by  means  of  a 
ouive.  Such  a  curve  often  gives  at  a  glance  information  which 
otherwise  could  be  obtained  only  by  considerable  computation ; 
and  in  many  cases  reveals  facts  of  peculiar  interest  and  impor- 
tance which  might  otherwise  escape  notice. 

The  use  of  graphic  methods  in  the  study  of  physics,  analytical 
mechanics,  and  engineering,  as  well  as  in  many  other  branches  of 
scientific  investigation,  is  already  extensive  and  is  rapidly  in- 
creasing. Graphic  methods  can  be  used,  however,  not  only  in 
examples  where  the  equation  connecting  the  two  variable  quan- 
tities is  known,  such  as  those  already  given,  but  also  in  examples 
where  no  such  relation  can  be  found ;  in  these  latter  cases  the 
graphic  method  furnishes  almost  the  only  practical  means  of 
studying  the  relations  involved. 

Comparative  statistics,  and  results  of  experiments  and  direct 
observations,  can  frequently  be  more  concisely  and  forcibly  rep- 
resented graphically  than  by  tabulating,  numerical  values.  The 
following  are  simple  examples  of  this  kind : 

1.  The  following  table  shows  the  net  gold  (to  the  nearest  million  of  dol- 
lars) in  the  U.  S.  Treasury  at  intervals  of  one  month,  from  Jan.  10,  1893 
to  Oct.  31, 1894  (Report  of  the  Sec.  of  the  Treas.,  1894,  p.  8): 


80 


LOCI    AND   THEIR   EQUATIONS. 


[23. 


Jan.  10.. 
Feb.  10.. 
Mar.  10.. 
Apr.  10.. 
May  10.. 
June  10.. 


Millions 
of  Dollars. 


120 
112 
102 
106 


1893. 


July  10. 
Aug.  10. 
Sept.  9. 
Oct.  10. 
Nov.  10- 
Dee.    9. 


Millions    il 
of  Dollars ' 


97 
103 


1894. 


Jan.  10. 
Feb.  10. 
Mar.  10. 
Apr.  10. 
May  10. 
June    9. 


Millions 
of  Dollars 

74 
104 
107 
106 

92 


Millions 
of  Dollars. 


July  10..  I 
Aug.  10. 
Sept.  10. 
Got.    10. 
Oct.   31. 


60 
61 


Using  time  (in  months)  as  abscissas,  and  dollars  (1,000,000  per  unit)  as 
ordinates,  the  separate  points  represented  by  the  table  have  been  plotted 
(Fig.  1)  and  then  joined  by  a  smooth  curve. 


Be= 

IQ9B3 

saiiJi 

"■ 

!^ 

OKH 

i^ 

ism 

■^ 

— ^ 

*■ 

lan 

"^ 

■m 

■"■ 

■■* 

■■■ 

■" 

% 

k. 

\ 

__ 

_^^ 

too 

,^^ 

\ 

-^ 

-s, 

r 

\ 

^ 

i 

\ 

V. 

^ 

I 

- 

1 

^ 

1 

\ 

v 

J 

\ 

^ 

\ 











1 

— 

V 

, 

■^ 

*= 



— 

"iO 

1 — * 

— 

- 

.., 

0 

1      2     3     4      5      6      7      8      9     10     11    12      1      2      3      4     5     6      7     8     9    10    1]     12 
1893                                                                         1894 

FIG.  1. 

In  this  example  the  curve  gives  no  ?ieiu  information,  but  it  presents  in  a 
much  more  concise  form  the  information  given  by  the  tabulated  numbers. 
Observe  also  that  if  the  points  arc  inaccurately  located,  the  diagram  be- 
comes not  only  worthless,  but  misleading. 

2.  An  excellent  example  of  the  use  and  advantages  of  the  graphic 
method  of  representing  comparative  statistics  is  found  in  the  large  engraved 
plate  placed  under  the  front  cover  of  the  Annual  Report  of  the  Secretary 
of  the  Treasury  for  1894.  This  plate  presents  on  a  single  sheet  informa- 
tion that  covers  several  pages  when  expressed  in  tabulated  numbers.  All 
of  the  curves  given  on  this  plate,  except  one,  are  shown  (on  a  smaller  scale) 
in  Fig.  2.  This  figure  should  be  carefully  studied,  and  if  possible  the 
original  plate  should  be  consulted. 

3.  The  curves  in  figures  1  and  2  were  constructed  by  locating  separate 
points  and  then  drawing  a  smooth  curve  through  these  isolated  points. 
Such  curves  give  no  new  information,  but  only  represent  graphically  in- 
formation already  ascertained. 

In  some  cases,  however,  curves  can  be  drawn  mechanically.  When  this 
is  possible  the  curve  is  constructed^  not  for  the  purpose  of  exhibiting  facts 
previously  known,  but  for  the  purpose  of  obtaining  new  information.  For 
instance,  in  the  stations  of  the  U.  S.  Weather  Bureau  an  instrument  called 


!S.J 


LOCI    AND    THEIR    EQUATIONS. 


31 


§        8         §         i         1         8         1 

"*  rK  "             rp" 

:ij>                                     '  '\\              II                             P^     i«    rf^     i"     N     r 

♦i     .W  \      ■        '        •! 

^  TT                   n-^-                   ^  f  9  ^  ^  o^ 

•1\T                 1'"-^-r^~"    r'     2  e  2  ^  1.  ^ 

^    ii-     \       -                    --:__                         Jtjjpnl^ci 

S     i                                          '^                                                "=£.S.5°*2- 

S    7'.\     ^             "        K^                       ~*                2    o     ?    i    ^    «»* 

;j: ^.t_  _..__^--±  ::_:":    s  -.  3.  a  ^  = 

s  I    l.-\  ::      .  -J        J:    £  :^  ?  3-  ?  s 

2  t  -,   -a   --            -C         .-         ^  H   S.  n  s*  s 

r  -"y    -r-              -r-..                  =   i  5:  5-  '•  ^ 

^  '\    i\  -              i                 =  -^  §  i-  = 

S  -i-     t  ••  %             i"                    ^           1  :;^ 

g             /      ••.        \                 i}                                                       a     2 

1    T^      ;     1              'S                                       ^ 

^  rH  ^^^3      ^^ ^ 

1^3    I   ^.tD       i 

S   T      5__   ^^    X               -^T    _ 

^     \                    \ 

^    i                   •    \                 .J 

22        1                                               1                 \                                *JS- 

"  I                •       \            '*v                   it 

!___:::  -z-^    5  __  :__^i: iTj:.:: 

H-  ;      '  -+-      ^L    x-           *            -t-  ' 

§     i        .V            _   ^co         ^t        -     -t-    A           -^                    -        - 

SI    _^._     .•• "       Sj_    Ti- 

:      T      •            ¥1            ^ 

^      '              4-         -^                                                    -i-     -r- 

5  L   -i   i        -i  -     ^i  -          

§  V       \    ^-            V      •£ 

;_     \    4^ it :--. 

^   \  ^.     -,-       --^-i-     -^s.!            -  --       --   --- 

£  ^  _t  ^        ^  i   -5^       i 

3-1        L      _      v     -  -4        -  '.- 

i         T              ••         t 

H-  -i        1^      -          ■*-      \             j^     --     - 

m      ^                  1                                                          *.         ^w                         ^--^ 

-;x:::aj:_::_:  ::_::::^-.X------ ,I------ 

^  i     jL    _        .  -__  ^y^  __  J  _  __- 

^    A            i                                       -       %^r       -    -i      - 

§    ^           L                                                I   S      ^-^ 

j::::L    :::_  i— _  : :-u^-a  ,« 

^               Jh                                               ^    ' 

0      "^                I                                                                                      *  S\'  -' 

J^      y              _______          J. 

^  ? ::::::::::::; ^v.":::: 

§     Zui  J    :::        :  :       c-~-    :i  -S  :    : 

±^.                                                            L          C    T        __ 

1           Z^  ^   «                      --"     •  1  IK 

g                 X    ^.'^*                  ~            fl                   t    M 

__:::. i.:;:^^::-  :  :    :  [_::-:— -"^-ji 

/     '^    ^           2""             t    ^^ 

5              "'\  ^     S"'^               t    S 

«                   \  "^    ,'■'      ~             i     1 

CO                                                                                 L^_                                                             -        ^__|^       _ 

®         /          '  '^th^    ---    ---__:  l-iff 

-  :::::::::;::i^i;E::....i. 1-1„- 

32 


LOCI    AND    THEIR    EQUATIONS. 


[23. 


the  Thermograph*  constructs  automatically  a  curve  which  shows  the  con- 
tinuous variation  of  the  local  temperature.  Similarly  the  Barograph* 
records  the  variation  of  the  barometric  pressure,  etc. 


Fig.  3.— Thermographs  for  Aug.  9-10  and  Sept.  27-28, 


Mon.  13 
XII 


Tu. 
XII 


Mt 


Wed. 

A^       - 

XII 


t.  27-28, 

1899, 

at  Lincoln, 

Neb 

Th. 

Fri.  17 

XII 

Mt 

XII 

Mt 

Fig.  4.— Barograph  Sheet,  March  13-17, 1899,  at  Lincoln,  Neb. 


Figures  3  and  4  are  copies  of  curves  thus  constructed  in  the  local  station 
at  Lincoln,  Neb.  The  upper  curve  in  Fig.  3  shows  the  temperature  from 
10  p.  M.  Aug.  8, 1899,  to  9  a.  m.  Aug.  11, 1899;  the  lower  from  11  p.  m.  Sept. 
26,  1899,  to  8  A.  M.  Sept.  29,  1899.  Interpret  these  curves.  Notice  espe- 
cially the  record  from  6  p.  m.  to  midnight  Aug.  10. 

The  varying  pressure  on  the  piston  in  the  cylinder  of  a  steam  engine  is 
determined  in  the  same  way  by  means  of  a  similar  instrument,  called  an 
Indicator.* 

4.  Exhibit  graphically  the  information  contained  in  the  following  table 
of  wind  velocities  for  Jan.  20  and  June  15  and  25,  1894: 

*  For  a  description  and  cut  of  the  "Thermograph,"  "Barograph,"  and  "  Indicator," 
see  these  words  in  the  Century,  Standard,  or  Webster's  International  Dictionary. 


26.] 


LOCI    AND   THEIR   EQUATIONS. 


Day. 

12-1 

1-2 

2-3 

3-4 

4-« 

5-6 

6-7 

7-8 

8-9 

9-10 

10-11 

11-12 

Jan.  20,  A.M... 
June  15,  A.  M... 
June25,  a.  M... 

3 
15 
17 

6 
11 
14 

7 
8 
13 

7 

10 
13 

8 
8 
11 

9 

9 

23 

12 
3 
23 

15 
3 
13 

15 
11 
9 

19 
16 
4 

12 
17 
2 

21 
21 

10 

Jan.  20,  P.M... 
June  15,  p.  M... 
June25,  p.  M... 

22 
15 
12 

22 
21 
15 

18 
22 
11 

19 
20 
12 

14 
17 
12 

9 
17 
5 

6 
12 

1 

7 
5 
3 

6 
5 
6 

6 
6 

7 

5 

6 

7 

4 
3 
3 

Intersection  of  Loci. 

24.  To  find  the  points  of  intersection  of  two  loci  when  their  equa- 
tions are  known. 

Since  the  points  of  intersection  of  two  loci  lie  on  both  curves, 
their  coordinates  must  satisfy  both  equations.  Therefore,  to  find 
the  coordinates  of  the  points  of  intersection  of  two  loci  we  treat 
their  equations  simultaneously,  regarding  the  coordinates  as  the 
unknown  quantities,  and  thus  find  the  values  of  the  coordinates 
which  satisfy  both  equations.  A  pair  of  values  which  satisfy 
both  equations  are  the  coordinates  of  a  point  of  intersection  of 
the  two  loci. 

If  the  equations  are  both  of  the  first  degree,  there  will  be  but 
one  pair  of  values  of  coordinates  satisfying  them,  and  therefore 
but  one  point  of  intersection  of  the  loci. 

If  one  or  both  of  the  equations  be  of  a  higher  degree  than  the 
first,  there  will  be  several  pairs  of  roots,  and  one  point  of  inter- 
section for  each  pair.  The  loci  will  then  have  several  points  of 
intersection. 

If  of  a  pair  of  roots  even  one  is  imaginary,  there  is  no  corre- 
sponding real  point  common  to  the  two  loci.  We  then  say  the 
intersection  is  imaginary. 

Since  imaginary  roots  of  equations  always  occur  in  pairs,  im- 
aginary intersections  of  loci  always  occur  in  pairs ;  and  hence 
the  passage  from  a  real  pair  of  intersections  to  an  imaginary  one 
Is  through  a  coincident  pair.  Suppose,  for  example,  a  straight 
line  intersects  a  circle  in  two  real  points.  If  the  line  be  moved 
so  that  it  becomes  tangent  to  the  circle,  the  two  points  of  inter- 
section coincide  in  the  point  of  contact.  If  the  line  be  moved 
still  farther,  the  intersections  are  said  to  become  imaginary. 

25.  Intercepts  on  the  axes  of  coordinates. 

This  is  a  special  and  very  important  case  of  the  preceding  sec- 
tion in  which  one  of  the  given  equations  is  x  =  0,  or  y  =  0. 


34  LOCI   AND   THEIR   EQUATIONS.  [26. 

To  find  the  points  of  intersection  of  a  curve  with  the  x-SbXXSy 
put  y  =  0  in  the  equation  of  the  curve  and  solve  the  resulting 
equation  for  x.  The  roots  of  this  equation  in  x  represent  the 
distances  from  the  origin  to  the  points  of  intersection ;  and  these 
distances  are  called  the  x-intercepts  of  the  given  curve. 

Similarly,  to  find  the  y-intercepts,  put  a?  =  0  in  the  given 
equation  and  solve  the  resulting  equation  for  y. 

Ex.  1.    How  many  x-intercepts  may  a  curve  of  the  nth  degree  have? 

Ex.  2.  What  does  it  mean  when  in  an  equation  in  polar  coordinates  we 
put^  =  0?    p  =  0? 

26.  A  line  may  be  defined  as  the  path  of  a  moving  point. 
Then,  if  (x,  y)  be  the  moving  point,  both  x  and  y  are  variable 
quantities,  and  are  called  the  variable  or  current  coordinates 
of  the  moving  point.  The  path  of  the  moving  point  is  then  de- 
termined by  the  condition  that  its  coordinates  must  vary  only  in 
such  a  manner  as  always  to  satisfy  a  given  equation ;  i.  e.  although 
both  coordinates  vary  the  relation  between  them,  remains  fixed. 

EXAMPLES. 
Find  the  intercepts  and  the  points  of  intersection  of  the  following  loci : 

1.  2x  +  Sy  =  12,        4x  — 2/  =  5. 

2.  Sx-\-by  =  i,        x  —  Sy  +  l  =  0. 

3.  5a:  — 22/  +  4  =  0,        10a:  —  4?/ +  3  =  0. 

4.  a: +  32/ =15,        x'+f=^25. 

5.  3x  — 4i/  =  20,        x''  +  if  —  10x—i0y-\-26  =  0. 

6.  5a:  +  42/  =  20,        x''  +  y''^4.. 

7.  x-3y  =  0,        x2  +  2,'2  +  20?/  =  0. 
S.    y"^  =  iax,        2xy  =  a^. 

9.    2/'  =  4aa;,        y''—x^  =  a\ 

10.  Find  the  points  of  intersection  of  the  loci  of  Nos.  1,  2,  3,  9,  15,  17, 
18,  19,  20,  21,  26  in  the  last  preceding  set  of  examples. 

11.  Find  the  intercepts  of  the  loci  of  Nos.  7,  9, 10,  11,  12,  13,  14,  18,  19, 
20  of  the  same  set  and  check  the  results  by  the  plots  already  made. 

12.  Find  the  area  of  the  triangle  whose  sides  are  x  —  3?/ +  5=0, 
3a:  +  42/  =  ll,    2x-\-ly  =  S, 

13.  What  is  the  area  of  the  quadrilateral  whose  sides  are  x  =  a,  y  =  b, 
bx  +  ay  =  0,    and    bx-\-ay  =  ab? 


27.] 


LOCI   AND   THEIR   EQUATIONS. 


35 


Symmetry  of  Loci. 

27.  The  process  of  constructing  a  locus  explained  in  §  21  is 
long  and  tedious.  It  may  often  be  shortened  by  an  examination 
of  the  peculiarities  of  the  given  equation,  such  as  the  limiting 
values  of  the  variables  for  which  both  are  real  (see  Ex.  3,  §  22), 
symmetry,  etc.  Such  considerations  will  often  reveal  the  general 
form  and  limits  of  the  curve  and  give  all  the  information  desired 
with  little  labor.  The  intercepts  (§  25)  are  almost  always  useful 
for  this  purpose. 

Definitions.  Two  points  A  and  B  are  said  to  be  symmetrical 
with  respect  to  a  centre  0  when  the  line  AB  is  bisected  by  0. 

Two  points  A  and  B  are  said  to  be  syirvmetrical  with  respect  to  an 
axis  when  the  line  AB  is  bisected  at  right  angles  by  the  axis. 

The  two  points  (Xj  y)  and  ( —  a?,  —  y)  are  symmetrical  with 
respect  to  the  origin  ;  (a?,  y)  and  (x.  —  y)  with  respect  to  the 
ic-axis.  / 

A  curve  is  said  to  be  sym- 
metrical with  respect  to  a  centre 
0  when  all  lines  passing 
through  0  meet  the  curve  in  a 
pair,  or  pairs,  of  points  sym- 
metrical with  respect  to  0. 

A  curve  is  said  to  be  sym- 
metrical with  respect  to  an  axis 
when  all  lines  perpendicular 
to  the  axis  meet  the  curve  in 
a  pair,  or  pairs,  of  points  sym- 
metrical with  respect  to  the 
axis. 

Or,  in  other  words,  a  curve  is  symmetrical  with  respect  to  an 
axis,  if  the  figure  appears  the  same  when  a  plane  mirror  is  placed 
on  the  axis  perpendicular  to  the  plane  of  the  curve. 

The  curve  PQ  is  symmetrical  with  respect  to  the  origin,  and 
RS  is  symmetrical  with  respect  to  the  t/-axis. 


36  LOCI   AND   THEIR   EQUATIONS.  [28c 

28.     Equations  in  Cartesian  Coordinates. 

(1)  If  f(x,  y)  =f(x,  —  2/)>*  ^^^  locus  of  the  equation 

Kx,y)=0 
is  symmetrical  with  reject  to  the  x-axis;  i.  e. 

Ij  an  equation  is  not  altered  when  the  sign  of  y  is  changed,  its  locus 
is  symmetrical  with  respect  to  the  x-axis. 

Let  (x'j  y')  be  any  point  on  the  locus /(^,  y)  =0. 
Then,  since  fi^x,  y)  ^f^x,  —  y),  by  hypothesis, 

That  is,  the  point  (x',  — y')  is  also  on  the  locus.  Therefore, 
if  the  line  x  =  x'  meets  the  locus  in  any  point  (x',  y'),  it  will 
also  meet  the  locus  in  the  symmetrical  point  {x',  —  ?/'),  and  the 
curve  is  symmetrical  with  respect  to  the  £c-axis. 

Ex.    Let  f{x,  y)=y^  —  4x,  then  /{x,  —y)  =  {—  yf  —  4x  =  ?/2  —  4x. 

Therefore  /(x,  y)  'Eifix,  —  y )  and  the  curve  y'^  —  4x  ==  0  is  symmetrical  with 
respect  to  the  x-axis.  (See  Ex.  2,  §  22.) 

(2)  Similarly,  if  fix,  y)=f( — x,  y)  the  locus  of 

Kx,y)=0 
is  symmetrical  with  respect  to  the  y-axis. 

Ex.  y  —  cosx^y  —  cos( — x). 

Therefore  the  locus  oi  y  =  cos  x  is  symmetrical  with  respect  to  the  y-axis, 

(3)  //  fix,  y)  =  ±  fi—  X,  —  y)  the  locus  of 

fix,y)=0 
is  symmetrical  with  respect  to  the  origin. 

Let  ix%  y')  be  any  point  on  the  locus  fix,  y)  =0. 

Then,  since /(a:;,  y)=  ±  fi — x,  — y)^J  hypothesis, 
Kx',y')=fi-x',-y')=0. 

Hence  the  straight  line  through  the  origin  and  the  point  ix',  y') 
meets  the  locus  again  in  the  symmetrical  point  ( — x',  — y'). 
Therefore  the  curve  is  symmetrical  with  respect  to  the  origin. 


-  {-xY  ,  {-yy 


1. 


*The  sign  "  =  "  means  *'  identical  with,"  t.  e.  the  same  for  all  values  of  x  and  y,  and 
Oierefore  that  the  two  expressions  vanish  for  the  same  values  of  x  and  y. 
E.g.     ia;  +  2/)-  =  a:2  +  2a;2/  +  j/2,    cos  x  =  cos  (— x-) . 


28.]  LOCI   AND   THEIR   EQUATIONS.  37 

Therefore  the  curve  — ,  -f  ^  =  1  is  symmetrical  with  respect  to  both  axes 
and  the  origin.  (See  Fig.  of  §  34.) 

(4)  If  S{x,  y)  =f(y,  x)  the  locus  of  J(x,  y)  ^0  is  symmetrical 
with  respect  to  the  bisector  of  the  angle  XOY. 

(5)  If  f{—x,  y)=f(—y,  x)  the  locus  of  f(x,  y)  =0  i^  sym- 
metrical with  respect  to  the  bisector  of  the  angle  X'  0  Y. 

Let  the  student  prove  propositions  (4)  and  (5). 

The  foregoing  conditions  of  symmetry  are  both  necessary  and 
sufficient;  i.  e.  if  either  one  of  the  conditions  (3),  for  example,  is 
satisfied,  the  locus  is  symmetrical  with  respect  to  the  origin, 
otherwise  not.  Let  the  student  examine  the  opposite  proposi- 
tions. 

The  following  conditions,  (6),  (7),  (8),  are  sufficient,  but  not 
necessary: 

(6)  If  an  equation  contains  only  even  powers  of  y,  its  locus  is  sym- 
metrical with  respect  to  the  x-axis. 

.   (7)     If  an  equation  contains  only  even  powers  of  x,  its  locus  is  sym- 
metrical with  respect  to  the  y-axis. 

(8)  If  an  equation  contains  only  even  powers  of  both  x  and  y,  its 
locos  is  symmetrical  with  respect  to  both  axes  and  also  with  respect  to  the 
origin. 

In  an  algebraic*  equation  either  one  of  the  following  conditions 
is  sufficient,  and  one  or  the  other  is  necessary. 

(9)  If  all  the  terms  of  an  algebraic  equation  are  of  even  degree,  or 
if  all  the  terms  are  of  odd  degree,  its  locus  is  symmetrical  with  respect  to 
the  origin. 

Show  that  (6),  (7),  (8),  and  (9)  follow  from  (1),  (2),  and  (3). 

Show  that  (6),  (7),  (8)  are  necessary  conditions  of  symmetry  if  the  equa- 
tion is  algebraic. 

♦  A  function  in  which  the  variables  are  involved  in  no  other  ways  than  by  addition,  sub- 
traction, multiplication,  division,  and  root  extraction  is  called  an  Algebraic  Function. 
All  others  are  called  Transcendental  Functions. 

ax"^  ■+■  bifi  

E.  g.    3x5  — 2x  -I-  4,    x-—axy  +  fty*.    ~  -r  n  ^xy^ 

are  algebraic  functions;  while  a^,  sin  x,  sec-^y,  log  ^x'"  +  y)  are  transcendental  functions. 


38  LOCI    AND   THEIR   EQUATIONS.  [29. 

29.     Equations  in  Polar  Coordinates. 

It  has  been  shown  in  the  first  chapter  that  for  all  values  of  0 

(p,—0)  =  (—p,  ^  —  0);    (/>,  7:-0)  =  (—p,  —6); 

(p,  71-^6)  =  (—p,  0), 

Prom  the  definitions  of  symmetry  it  follows  that,  for  all  values 
of  0, 

(Py  6)  and  (p,  — 6),  or  (/>,  t:  —  d)  are  symmetrical  with  re- 
spect to  OX'j 

(jO,  0)  and  {p,  TT  —  0),  or  ( — py  — 0)  are  symmetrical  with  re- 
spect to  OF;* 

(jO,  0)  and  (p,  t:  -\-6^j  or  ( — />,  0)  are  symmetrical  with  respect 
toO. 

Also  from  the  definition  of  symmetrical  curves  we  may  say 
that  a  curve  is  symmetrical  with  respect  to  a  centre,  or  an  axis,  when 
every  point  on  the  curve  has  its  symmetrical  point  with  respect  to  the 
center,  or  axis,  on  the  curve. 

Hence  the  following  are  sufficient  conditions  of  symmetry  for 
loci  in  polar  coordinates : 

(1)  If  m^f{—0),  or,  if  f(d)=—f(n  —  0)y  the  locus  oj 
p  =zf(^d)  is  symmetrical  with  respect  to  OX. 

For,  in  the  first  case;  the  value  of  p  is  the  same  when  0  =  0'  a>Q 
vrhen  0=:  —  0']  and  in  the  second  case,  the  values  of  p  corre- 
sponding to  0  =  0'  and  0  =z-!T  —  0'  differ  only  in  sign. 

Hence  in  either  case,  if  any  point  {p',  0')  is  on  the  locus,  its 
symmetrical  point  (/>',  — 0')  is  also  on  the  locus  ;  therefore  the 
locus  is  symmetrical  with  respect  to  OX. 

(2)  SimUarly,  if  f{0)  -/(tt  —  ^),  or,  if  f(0)=—f(^0),  the 
locvis  of  p:=:f(^0)  is  Symmetrical  with  respect  to  OY. 

(3)  If  f(^0)  =/(7r  +  0),  the  locus  of  p  =:f(0)  is  symmetrical  with 
respect  to  0. 

(4)  The  locus  of  p'^=f(0')  is  symmetrical  with  respect  to  0. 
E.  g.    The  locus  p  =  cos  ^  is  symmetrical  with  respect  to  OX. 

The  locus  p  =  2a  sin  0  is  symmetrical  with  respect  to  OY.    (See  Ex.  4, 

§22.)       

*  OF  is  assumed  perpendicular  to  the  initial  line  OX. 


29.]  LOCI   AND   THEIR   EQUATIONS.  39 

The  locus  p  =  sin  2^  is  symmetrical  with  respect  to  OX,  0  F,  and  O.  (See 
Ex.  5,  §  22.) 

Show  that 

(5)  if/(45°+<?)=/(45°  — ^),  the  loms  of  p  =/{$)  is 
rical  with  respect  to  the  line  0=z  4:5°. 

(6)  ijr /(135°  +  0)  3/(135°  —  0),  the  locus  of  />  =f(0)  is 
metrical  with  respect  to  the  line  ^  =  135°. 

Are  the  conditions  (5)  and  (6)  satisfied  by  the  equation  of  the  locus  shown 
in  Ex.  5,  §  22  ? 

EXAMPLES. 
In  what  respects  are  the  loci  of  the  following  equations  symmetrical  ? 
1,'  y  =  x\  2.    y^  =  x. 

3.    y=2?.  4.    3/2  =  a^. 

6.  i/2  =  a;2.  6.    2/2  =  x*. 

7.  2/'=a^-  8-    y^=x\ 

9.  y  =  3i?  —  X,  10.  y  =  a:*  —  x^, 

11.  xy=^a.  12.  ax2  +  6/  =  l. 

13.  ax'-{-2hxy-{-cy'^  =  \,  14.  oar*  +  26a;y  +  ay^  =  1. 

15.  axy-{-h{x-\-y)  =  c,  16.  x^-f-y^=l«    ' 

17.  x*+y=l.  18.  ic*=y2(4ct2_a.2), 

19.  x(^  +  x)2  +  aV  =  0.  20.  a;y  =  a2(x2+y2)^ 

^21.  xy2-|-4(a;+y)  =  0.  22.  x«+^%  =  a«. 

23.  (a  — a;)2/2  =  (a  +  a5)a;2,  24.  {a  —  x)y''-\-2^  =  0. 

25.  p2^sin2^.  26.  /o2  =  cos2/?. 

27.  Point  out  the  symmetric  properties  of  the  loci  in  the  last  two  pre- 
ceding sets  of  examples,  especially  those  given  in  polar  coordinates. 
Check  the  results  by  referring  to  the  plots  already  made. 

28.  Show  that  if  an  equation  is  not  altered  when  —  x  is  written  in  the 
place  of  2/,  and  y  in  the  place  of  x,  its  locus  will  show  no  change  in  posi- 
tion when  the  curve  is  turned  about  the  origin  through  a  right  angle  in  its 
plane. 

For  example,  the  locus  of  the  equation 

X*  +  o^^^y  —  2/*  =  0 
is  such  a  curve. 


40 


LOCI   AND   THEIR   EQUATIONS. 


[30. 


To  Find  the  Equation  of  a  Locus,  having  given  its  Geo- 
metric Definition. 

30.  It  should  be  borne  in  mind  that  to  find  the  equation  of  a 
locus  we  have  merely  to  find  an  equation  that  is  satisfied  by  the 
coordinates  of  every  point  on  the  locus,  and  not  satisfied  by  the 
coordinates  of  any  other  point.  It  is  not  easy  to  give  specific 
directions  which  can  be  applied  in  all  cases,  but  the  following 
plan  will  be  useful  to  the  beginner,  at  least  in  the  simpler  cases : 

(1)  Choose  the  system  of  coordinates  best  adapted  to  the 
locus  under  consideration,  and  select  a  convenient  set  of  axes. 

(2)  Write  down  the  geometric  equation  which  expresses  the 
given  geometric  definition,  or  any  known  geometric  property  of 
the  locus. 

(3)  Express  this  geometric  equation  in  terms  of  the  chosen 
system  of  coordinates,  and  simplify  the  result. 

The  following  examples  will  give  a  better  idea  of  the  method 
of  procedure  than  any  formal  rules ;  they  should  be  carefully 
studied : 

31.  To  find  the  equation  of  any  straight  line. 


B 

Y 

I 

R 

A^X*^**^ 

^ 

0                                            < 

^ 

Let  ABC  be  any  straight  line  meeting  the  axes  in  A  and  B, 

Let  OB  =  b,  let  tan  XA  C  =  m. 

Let  P{x,  y)  be  any  point  on  the  line. 

Draw  PQ  parallel  to  OY,  and  BE  parallel  to  OX. 


31.]  LOCI    AND    THEIR    EQUATIONS.  41 

Then  for  the  geometric  equation  we  have 

QP=QR^RP=OB-\-  BR  tan  PER. 

But        qP  =  y,     OB  =  b,     BR  =  x,     tan  PBR  =  m. 

.'.     7jz=mx-\-b,  (1) 

which  is  the  required  equation. 

For  any  particular  straight  line  the  quantities  m  and  b  remain 
the  same,  and  are  therefore  called  constants.  Of  these,  m,  the 
tangent  of  the  angle  between  the  line  and  the  £c-axis,  is  called 
the  Slope  of  the  line,  while  b  is  the  ^/-intercept. 

By  giving  suitable  values  to  the  constants  m  and  6,  (1)  may  be 
made  to  represent  any  straight  line  whatever,  e.  g. 

If  6  =  0,  we  have 

y  =  mx,  ,  (2) 

for  the  equation  of  any  line  through  the  origin. 

Quantities  entering  into  an  equation,  such  as  m  and  &,  which 
remain  constant  so  long  as  we  consider  any  particular  curve,  but 
whose  variation  causes  a  change  in  the  position,  size,  or  shape  of 
the  curve,  are  called  Parameters  of  the  curve.  * 

Moreover,  any  equation  that  can  be  put  in  the  form  (1),  i,  e. 
y  equals  some  multiple  of  x  plus  a  constant,  represents  a  straight  line. 

The  general  equation  of  the  first  degree 

Ax-^By-^C=0  (3) 

J  C 

may  be  written  ?/  =  —  i^  —  r> 

and  therefore  (3)  represents  a  straight  line  whose  slope  is  —  ^ 

C 
and  whose  ;y-intercept  is  —  ^.    (See  §  43. ) 

/KEx.  1.  If  6  varies  in  (i)  while  w  remains  constant,  how  will  the  line 
enange  position?  If  m  varies  while  b  remains  constant?  If  m  varies 
in  (2)? 

Ex.  2.  What  will  be  true  of  the  signs  of  m  and  6  when  the  line  crosses 
the  various  quadrants  ? 

♦The  difference  between  parameters  and  coordinates  should  be  carefully  noted;  also 
the  difference  in  the  effect  of  a  variation  of  the  parameters  of  an  equation  and  the  varia- 
tion of  the  current  coordinates.    (See  §  26.) 


42 


LOCI   AND   THEIR   EQUATIONS. 


[32. 


32.     To  find  the  equation  of  a  circle  referred  to  any  rectangular  axes. 


Let  r  =  radius,  and  let  C{a,  b)  be  the  centre. 
Let  P(x,  y)  be  any  point  on  the  circle. 
Then  CP=  r.         [Geometric  equation.] 

But  CP^=  (x-ay  +  (y-by,  [(2),  §  7.] 

...     ^^_ay-\-(iy  —  by==r' 
is  the  required  equation. 

If  a  =  r  and  6=0,  (1)  reduces  to 

x'-{-y'  —  2rx  =  0. 
If  a  =  —  r  and  6  =  0,  (1)  becomes 

x'-^y'  +  2rx  =  0. 

The  circle  at  the  right  in 
the  figure  is  the  locus  of 
equation  (2);  the  circle  at 
the  left  is  the  locus  of  equa- 
tion (3). 

When  the  centre  is  at 
the  origin,  a  =  6  =  0,  and 
we  have  for  the  simplest 
equation  of  the  circle  in 
Cartesian  coordinates  the 
standard  form  (§  16^, 


(1) 


(2) 


(3) 


oc"^  -\-  y^  :=r^ 


(4) 


Ex.  1.    What  is  the  form  of  the  equation  and  the  position  of  the  circle, 
if  6=  =branda  =  0? 


34.] 


LOCI    AND   THEIR   EQUATIONS. 


43 


Ex.  2.  What  are  the  parameters  in  these  equations  ?  ^Discuss  the  effect 
produced  by  their  variation. 

Ex.  3.    Find  the  general  equation  of  a  circle  which  touches  both  axes. 

33.     Polar  equations  of  the  circle. 

It  follows  from  (1),  §  8,  that  the  polar  equation  of  the  circle 
whose  centre  is  at  the  point  (a,  «)  and  whose  radius  is  r,  is 

P^'—  2ap  cos  (^  —  a)  +  a'  —  7-2  =  0.  (1) 

If  the  pole  is  on  the  circle,  the  equation  is 

P  =  2rcos  (0  —  a)'j  (2) 

if  the  centre  is  also  on  the  initial  line,  the  equation  is 

P  =  2r  cos  0;  (3) 

if  the  circle  is  above  the  initial  and  tangent  to  it  at  the  pole,  its 
equation  is 

P—2rsm0.  (4) 

Ex.  1.  Why  is  (1)  of  the  second  degree  in  p  while  (2),  (3),  and  (4)  are  of 
the  first  degree  ?  When  is  the  pole  outside,  and  when  inside  the  circle  ? 
Discuss  the  effect  of  the  variation  of  the  parameters  in  these  polar  equa- 
tions. 

Ex.  2.    Transform  equations  (1),  (2),  (3),  (4)  to  rectangular  coordinates. 


34.  The  Ellipse.  The  ellipse  is  the  locus  of  a  point  which  moves 
80  that  the  sum  of  its  distances  from  two  fixed  points,  called  foci,  is  con- 
stant. 


Take  the  line  through  the  foci  as  the  a>axis,  and  the  point  mid- 
way between  the  foci  as  origin. 


44  LOCI    AND    THEIR    EQUATIONS.  [35, 

Let  2a  =  the  sum  of  the  distances  from  any  point  on  the  ellipse 
to  the  foci. 

Let  -F(c,  0)  and  F'(—  c,  0)  be  the  two  foci. 
Let  P(x,  y)  be  any  point  on  the  locus. 

Then  FP  +  F'P  =  2a.  [Geometric  equation.  ] 

But  FP  =  t/(.t  — c)2  +  2/', 


and  F'P=y  (x-\-Gy—y\  [(2),  §  7.] 


.-.     V{x-cy-\-f^V(x-i-cy^f  =  2a.  (1) 

Transposing  the  first  radical  and  squaring 


(x  +  cf  +  f  =  ^o?  +  (^  —  c)'  +  7/2  _  4a  V  {x  —  cy  -h  y' 


or  aV  (x  —  cy  -\-  y^  =z  a^  —  ex. 

Squaring  and  transposing  again 

(a^  —  c')x'  +  ahf  =  aXa'  —  c^. 

If  we  put  a^  —  c^  =  ¥,  we  get  the  equation  of  the  ellipse  in  the 
standard  form, 

35.  An  examination  of  this  equation  (2)  as  to  symmetry, 
limiting  values  of  the  variables  and  intercepts,  will  give  the  gen- 
eral form  and  limits  of  the  curve. 

(1)  Only  the  squares  of  the  variables  x  and  y  appear  in  this 
equation. 

Therefore  the  ellipse  is  symmetrical  with  respect  to  both  axes, 
and  also  with  respect  to  the  origin.    [(8),  §  28.] 

Hence  every  chord  passing  through  0  is  bisected  by  0.  For 
this  reason,  the  point  0  is  called  the  Centre  of  the  ellipse. 
Likewise  the  lines  A  A'  and  BB'  are  called  the  Major  Axis  and 
Minor  Axis,  respectively. 

(2)  When     y'=0,     x=zta,     ^-intercepts. 

When     x=  0,     y=±b,     7/-intercepts. 

Therefore  the  curve  cuts  the  x-Sixis  a  units  to  the  right  and  a 
units  to  the  left,  the  ^/-axis  b  units  above  and  b  units  below  tbe 
origin. 


36.]  LOCI   AND   THEIR   EQUATIONS.  45 

(3)     Solving  the  equation  (2)  for  y  and  x  respectively  we  find 
h 


y=  ±  -Va^  —  x\  x=  ±  J- V b'  —  y\ 

Hence  y  is  imaginary  when  ic  >  a,  or  a?  <  —  a  ;  and  x  is  imagi- 
nary when  yy-b,  or  ?/  <  —  b. 

Therefore  the  curve  lies  wholly  within  the  rectangle  formed  by 
the  lines  x^=  ±a  and  y=ztb. 

Also,  as  either  variable  increases,  the  other  diminishes.  The 
form  of  the  curve  is  shown  in  the  figure. 

Such  an  examination  of  an  equation  is  called  A  Discussion 
of  the  Equation. 

Ex,  1.  Transform  equation  (2),  §  34,  to  polar  coordinates  and  show  that 
p4s  finite  for  all  values  of  6. 

Ex.2.    Whereisthe  point  (/i,fc)  if  ^-f^^  —  l>0?    <0? 

Ex.^.  Show  the  relation  of  the  ellipse  —  -j-  ^  =  1  to  the  circles 
aj»+y2^a2  and  x"  -^y^  =  b-. 

36.  The  Hyperbola.  The  hyperbola  is  the  locus  of  a  point  which 
moves  80  that  the  difference  of  its  distances  from  two  fixed  points  (foci) 
is  constant. 

Choose  axes  as  in  the  case  of  the  ellipse,  let  2a  be  the  constant 
difference,  and  show  that  when  b'^  =  c^  —  a^  the  equation  of  the 
hyperbola  reduces  to  the  standard  form 

a'       b'  ^^ 

Ex.  1.    Discuss  equation  (1). 

Ex.  2.    Show  that  the  hyperbola  (1)  lies  wholly  between  the  two  straight 

lines  v  =  ±  —X.  and  that  as  x  becomes  infinite  the  ordinates  of  the  lines 

become  equal  to  the  ordinates  of  the  hyperbola.    These  lines  are  called 
the  Asymptotes*  of  the  hyperbola. 

Ex.  3.    Transform  equation  (1)  to  polar  coordinates,  and  find  the  value 

of  p  when  ^  =  d=  tan-^  — . 
a 

*  See  note  under  §  116. 


46 


LOCI    AND    THEIR    EQUATIONS. 


[38. 


37.  The  Parabola.  The  parabola  is  the  locus  of  a  pohit  whose 
distance  from  a  fixed  straight  line  is  equal  to  its  distance  from  a  fixed 
point. 

The  fixed  point  is  called  the  Focus;  the  fixed  line  the  Di- 
rectrix. 

Take  the  line  through  the  focus  perpendicular  to  the  directrix 
as  the  ic-axis,  and  the  origin  midway  between  the  focus  and  the 
directrix;  let  2a  denote  the  distance  from  the  focus  to  the  di- 
rectrix. 

Then  show  that  the  equation  of  the  parabola  is 


y^  =:  4:ax. 


(1) 


Discuss  this  equation  (1)  (see  Ex.  2,  §  22) ;  also  y^  =  —  4ax  and  x^  =  ±  4ay» 


Strophoid. 


Hence  in  the  triangle  AOQ 

I  OAQ  =  90°—  20,     and 


38.*  YOX  is  a  right 
angle  and  J.  is  a  fixed  point 
in  OX;  AG  is  any  line 
through  A  cutting  OF  in 
B;  P  and  Q  are  two  points 
on  AB  such  that 

QB  =  BF=  OB. 
The  locus  of  the  points  P 
and  Q  8iS  AG  turns  about 
A  is  called  the  Strophoid. 

To  find  the  polar  equa- 
tion of  the  Strophoid,  take 
the  point  0  as  the  pole  and 
OX  as  the  initial  line. 

Let  OA  =  a. 

Let  Q(p,  0)  be  any  point 
on  the  locus. 

In  the  isosceles  triangle 

Boq 

lBOQ=Z.  BQO=m''—e, 

and      lOBq-=2d. 


Z  0§A  =  90°-f  ^; 


39.]  LOCI   AND   THEIR  EQUATIONS.  47 

P       sin  (90°  —  20) 


whence 


a        sin  (90°  +  0) 
a  cos  20 


(1) 


cos  0    ' 
which  is  the  required  equation. 

If  we  multiply  (1)  by  /o^  it  may  be  written 

p\p  cos  0)  —  ap\co^^  0  —  sin'  0)  =  0.  (2) 

If  OX  and  0  F  be  taken  as  axes, 

p^=zo(^-\-'ifjPC06  0  =  x,     p  sin  0  =  y.      [(1),  §  6.] 

Substituting  in  (2)  gives  the  rectangular  equation 

x(ix'^y')-a(x'-f)=0.  (3) 

Ex.  1.    Show  that  the  coordinates  of  P  also  satisfy  equations  (1)  and  (3). 

Ex.  2.  Trace  the  change  in  the  values  of  f)  as  6  varies  from  0  to  tt  in 
equation  (1). 

Ex.  3.    From  a  discussion  of  equation  (3)  show  that: 

(1)  The  curve  is  symmetrical  with  respect  to  the  aj-axis. 

(2)  The  curve  cuts  both  axes  twice  at  the  origin  and  the  x-axis  once  at 
the  point  (a,  0). 

(3)  The  curve  lies  wholly  between  the  two  lines  x  =  ±:a. 

(4)  For  all  values  of  x  between  +  a  and  —  a,  y  is  finite,  but  for  x  =  —  a, 
p  is  infinite.  Therefore  the  curve  has  infinite  branches  in  the  second  and 
third  quadrants. 

The  Eose  of  Four  Branches. 

39.*  Given  a  line  AB  of  constant  length  (2a)  whose  extremi- 
ties are  free  to  move  along  two  perpendicular  lines  OX  and  0  Y. 
Find  the  locus  of  P,  the  foot  of  the  perpendicular  drawn  from  0 
to  AB. 

Take  0  as  the  pole,  and  OX  as  the  initial  line. 

Let  P(p,  0)  be  any  point  on  the  locus. 

Then  from  the  right  triangles  OPB  and  OAB^ 

P=  OB  cos  0, 
and  OB  =  AB  sin  OAB  =  2a  sin  0, 

.-.     />:=asin2^  (1) 

is  the  polar  equation  of  the  locus.     (See  Ex.  5,  §  22. ) 


48 


LOCI   AND    THEIR   EQUATIONS, 


[39. 


(2) 


Ex.  1.    Show  that  the  equation  in  rectangular  coordinates  is 

Ex.  2.     Find  from  equations  (1)  and  (2): 
(1)    The  smallest  circle  enclosing  the  curve. 

The  number  of  times  the  curve  passes  through  O,  as  d  varies  from  0 


(2) 

to27r 

(3) 
(4) 


The  different  sorts  of  symmetry. 

Where  the  point  (xi,  yi)  may  be  if  (xi^  -f  y^'^f  —  iax^yi^  <  0. 


Also  trace  the  curve  directly  as  it  is  generated  by  the  moving  line  AB. 

EXAMPLES. 

(l>)  A  moving  point  is  always  four  times  as  far  from  the  x-axis  as  from 
the  y-axis.    What  is  the  equation  of  its  locus? 

r'^  Find  the  locus  of  a  point  which  is  equidistant  from  the  two  points 
(3;  2)  and  (—  2,  1).  Ans.    5x  +  ?/  =  4. 

Find  the  locus  of  a  point  which  is  equidistant  from  the  points  (a,  6) 


and  (c,  d). 

^4./  A  point  moves  so  that  its  distance  from  the  point  (3,  — 4)  is  always  5. 
Find  the  equation  of  its  locus.  Does  the  locus  pass  through  the  origin? 
Why?  .  Ans.    x'  +  y'  —  6x  +  8y  =  0. 

(5*  Find  the  equation  of  a  circle  touching  both  axes  and  having  its 
centre  at  the  point  (—  3,  3). 

{6/  Find  the  equation  of  a  circle  touching  both  axes  and  having  a  radius 
equal  to  4. 


7. 


A  point  P  is  two  units  from  a  circle  with  radius  4  and  centre  at 
What  is  the  locus  of  P? 


[( s!    A  point  moves  so  that  its  distance  from  the  origin  is  twice  its  distance 
from  the  x-axis.    What  is  the  equation  of  its  locus?      Ans.    x^  —  3^  =  0. 


39.]  LOCI   AND   THEIR   EQUATIONS.  49 

^(9/  A  point  moves  so  that  its  distance  from  the  x-axis  is  equal  to  its  dis- 
tance from  the  point  (2, — 3).  Show  that  the  equation  of  its  locus  is 
a;^-4x  + 6^^  +  13  =  0. 

10/  A  point  P  moves  so  that  its  distances  from  the  points  A(2,  2)  and 
B^ —  2,  —  2)  satisfy  the  condition  AP-\-  BP  =  8.  Show  that  the  equation  of 
its  locus  is  3x2  _  2xy  -f  Zy'  =  32. 

,11;  What  is  the  locus  of  a  point  which  moves  so  that  (1)  the  sum,  (2) 
the  difference,  (3)  the  product,  (4)  the  quotient  of  its  distances  from  the 
axes  is  constant  (a)  ? 

12.  What  is  the  locus  of  a  point  which  moves  so  that  (1)  the  sum,  (2) 
the  difference,  (3)  the  product,  (4)  the  quotient  of  the  squares  of  its  dis- 
tances from  the  axes  is  constant  (a'^)? 

^3,'  Find  the  locus  of  a  point  which  moves  so  that  the  sum  of  the 
sqiiares  of  its  distances  from  the  points  (a,  0)  and  ( — a,  0)  is  constant  (20^). 

(f^.  Find  the  locus  of  a  point  which  moves  so  that  the  sum  of  the 
sqViares  of  its  distances  from  the  three  points  (5,  — 1),  (3,  4),  ( — 2,  — 3)  is 
always  64. 

i^.  Show  that  the  locus  of  a  point,  the  sum  of  the  squares  of  whose 
distances  from  n  fixed  points  is  constant,  is  a  circle. 

16.  Find  the  locus  of  a  point  which  moves  so  that  the  difference  of  the 
squares  of  its  distances  from  (a,  0)  and  ( — a,  0)  is  the  constant  20^. 

17.  Find  the  locus  of  a  point  such  that  the  sum  of  the  squares  of  its 
distances  from  the  sides  of  a  square  is  constant. 

18.  Find  the  locus  of  the  centre  of  a  variable  circle  which  touches  a 
fixed  circle  and  a  fixed  straight  line. 

19.  Find  the  locus  of  the  centre  of  a  circle  which  touches  two  fixed  cir- 
cles. Four  cases  should  be  considered.  What  does  the  locus  become  when 
the  fixed  circles  are  equal  ? 

20.  Find  the  locus  of  the  middle  points  of  all  chords  of  a  given  circle 
which  pass  through  a  fixed  point.  [Take  the  fixed  point  as  pole,  and  use 
the  polar  equation  of  the  given  circle.] 

21.  A  straight  rod  moves  so  that  its  ends  constantly  touch  two  fixed  per- 
peudicular  rods.    Find  the  locus  of  any  point  P  on  the  moving  rod. 

22y   On  a  level  plain  the  crack  of  a  rifle  and  the  thud  of  the  ball  striking 


the  target  are  heard  at  the  same  instant.    Find  the  locus  of  the  hearer. 
[S.  L.  Loney's  Coordinate  Geometry,  p.  283.] 

23).    In  a  given  circle  let  AOB  be  a  fixed  diameter,  OC  any  radius,  CD  the 
per{)endicular  from  C  on  AB;  let  P  and  Q  be  two  points  on  the  line  through 
O  and  C  such  that  Q,0  =  0P=  CD.    Find  the  locus  of  P  and  Q  as  OC  turns 
about  O. 
5 


50 


LOCI   AND  THEIR  EQUATIONS. 


[39. 


24.  A  and  B  are  two  fixed  points,  and  P  moves  so  that  PA  =  nPB. 
Find  the  locus  of  P. 

'25.  ^05  and  OO-D  are  two  straight  lines  which  bisect  one  another  at 
right  angles.    Find  the  locus  of  a  point  P  such  that  PA  PB  =  PC  •  PD. 

26.  If  ABC  is  an  equilateral  triangle,  find  the  locus  of  a  point  P  such 
thatPA  =  PB-\-PC. 

27.  AB  is  a  fixed  diameter  of  a  given  circle  and  AC  is  any  chord;  Pand 
Q  are  two  points  on  the  line  AC  such  that  QC^^  CP=  CB.  Find  the  locus 
of  P  and  Q  as  ^O  turns  about  A. 

28.  Any  straight  line  is  drawn  from  a  fixed  point  O  meeting  a  fixed 
straight  line  in  P,  and  a  point  Q  is  taken  in  this  line  such  that  OP '  OQ  is 
constant.    Find  the  locus  of  Q. 

29.  Any  straight  line  is  drawn  from  a  fixed  point  O  meeting  a  fixed  cir- 
cle in  P,  and  on  this  line  a  point  Q  is  taken  such  that  OP-  OQ  is  constant. 
Show  that  the  locus  of  Q  is  a  circle.     [See  suggestion  under  Ex.  20.] 


30.    The  Cissoid  of  Diodes.* 


Let  OX  be  a  fixed  diameter  of  a  given 
circle  and  CX  a  tangent  line ;  let  OA  be  any 
line  through  O  meeting  CX  in  A  and  the 
circle  in  B;  on  this  line  take  OP=BA. 
Then  the  locus  of  P  as  OA  revolves  about  O 
is  the  Cissoid. 

Using  OX  and  Oy  as  axes  show  that  the 
equations  of  the  Cissoid  are 
„  sin^^ 


and 


r 


2r  — x' 


where  r  is  the  radius  of  the  given  circle. 


31.    The  Ldmacon  of  Pascal.* 

Through  a  fixed  point  O  on  a  given  circle 
draw  any  secant  cutting  the  circle  again  in 
A;  and  on  this  line  take  QA=^AP=b,  a 
constant.  The  locus  of  the  points  P  and  Q 
as  OA  turns  about  O  is  called  the  Limacon. 

Show  that  the  equations  of  the  Limacon 
referred  to  OX  and  OF  as  axes  are 

p  =  a  cos  d±bf 
and         (x"  +  y2  —  axf  =  ^^(a;^  +  y^). 

Notice  the  three  different  forms  of  the 
curve  according  as  6  > ,  =,  or  <  a,  the  di- 
ameter of  the  circle. 


See  note  on  page  51. 


39.] 


LOCI    AND   THEIR    EQUATIONS. 


51 


32.    The  Conchoid  of  Nicomedes,* 

Through  a  fixed  point  O  draw  any  secant 
meeting  a  fixed  straight  line  AB  in  C,  and 
on  this  secant  lay  off  QC  =  CP=bf  a  con- 
stant. The  locus  of  the  points  P  and  Q  as 
OC  revolves  about  O  is  called  the  Oonchoid. 

Take  OX,  the  perpendicular  to  AB,  as 
initial  line  and  x-axis,  and  show  that  the 
equations  of  the  Conchoid  are 
P  =  a  sec  d±:b, 
and  (x^  +  ?/2)  (a;  —  a)^  =  b'^x\ 

Consider  the  fonns  of  the  curve  when 
6  > ,  =,  and  <  a,  where  a  =  OD. 


.  *For  a  historic  account  of  the  invention  of  the  Cissoid  and  Conchoid,  and  the  cause 
which  led  to  their  invention,  see  Cajori,  "A  History  of  Mathematics,"  1894,  p.  50,  and  Ball, 
"A  Short  History  of  Mathematics,"  1888,  p.  78. 

For  the  reason  why  the  ancient  geometers  desired  to  Duplicate  the  Cube,  see  Cajori, 
p.  25,  and  Ball,  pp.  38  and  75.  For  biography  of  Pascal,  see  Cajori,  pp.  175-77,  and  Ball, 
pp.  249-56. 


CHAPTER  III. 
THE  STRAIGHT  LINE. 

40.  It  was  shown  in  §  31  that  the  equation  of  any  straight 
line  when  expressed  in  terms  of  its  slope  m  and  ^/-intercept  b  is 
an  equation  of  the  first  degree, 

y  =  mx-\-b;  (1) 

and  also  that  the  general  equation  of  the  first  degree, 

Ax-\-By-\-C  =  0, 
represents  a  straight  line. 

It  is  sometimes  more  convenient,  however,  to  write  the  equa- 
tion of  the  straight  line  in  other  forms;  i.  e.,  to  express  it  in 
terms  of  some  other  pair  of  parameters. 

41.  To  find  the  equation  of  the  straight  line  in  terms  of  its  inter- 
cepts on  the  axes. 


Let  J.  and  5  be  the  points  in  which  the  straight  line  meets  the 
axes  and  let  OA  =  a,  and  OB  =i). 
Let  P{x,  y)  be  any  point  on  the  line. 
Draw  PQ  parallel  to  the  ^/-axis,  and  join  0  and  P. 
Then  A  OAP  +  A  OBP  =  A  OAB. 

HencG  bx  -\-  ay  =  ab. 


or 


o^  b 


(1) 


42.] 


THE   STRAIGHT   LINE. 


53 


If  ^  =  —  and  m=-f,  the  equation  may  be  written 
Ix  -\-my  =  1. 


(2) 


42.  To  find  the  equation  of  a  straight  line  in  terms  of  the  length  of 
the  perpendicular  from  the  origin  upon  the  line  and  the  angle  which  that 
perpendicular  makes  with  the  x-axis. 


Let  ONhQ  perpendicular  to  the  straight  line  AB,  and  intersect 
it  in  R. 
"  Let  OR  =p,  and  angle  XON=a. 

Let  P(a?,  y)  be  any  point  on  the  line. 

Then  since  OQPR  is  a  closed  polygon,  OR  is  equal  to  the  sum 
of  the  projections  of  OQ,  QP,  and  PR  upon  OR,     That  is, 

OR  =  proj.  of  OQ  +  proj.  of  QP  +  proj.  of  PR. 

=  OQ  cos  a  +  QP  sin  a  +  0. 

.*.     X  cos  a  +  y  sin  a  =Pj  (1) 

which  is  the  required  equation. 
Let  angle  XAP  =  ^  =  90°  +  «. 

Then  cos  a  =  sin  y,     sin  a  =  —  cos  ^, 

and,  by  substituting  in  (1),  the  equation  of  the  line  becomes 

ic  sin  ^  —  y  cos  y  =p.  (2) 

Since  equations  (1)  and  (2)  involve  the  trigonometric  func- 
tions, sin  and  cos,  ON  and  AB  must  be  regarded  as  directed  lines. 
As  in  Trigonometry,  we  will  consider  the  directions  of  the  termi- 
nal lines  of  a  and  y  as  the  positive  directions  of  these  lines. 

If  ^  =  90°+ a,  as  assumed  above,  then  standing  at  jR  facing 
the  positive  direction  of  ON,  the  positive  direction  of  AB  is  to 


64 


THE   STRAIGHT   LINE. 


[42. 


the  left;  and  standing  at  R  facing  the  positive  direction  of  AB^ 
the  positive  direction  of  ON  is  from  AB  toward  the  right. 

This  will  be  called  the  positive  side^  of  the  litie  AB. 

Then  in  equations  (1)  and  (2)  _p  is  positive  when  taken  in  the 
positive  direction  of  ON.  Hence  when  p  is  positive  the  origin  is 
on  the  negative  side  of  the  line. 


E.  g.    In  the  equations 
±3, 


i/2"^V2 


.  • .    «  =  45°  and  y 
for  both  lines ;  but 
for  AB  p  =  3, 

forOD  p  =  —  3. 

Hence  the  two  lines  are  parallel 
but  on  opposite  sides  of  O.  Also 
O  is  on  the  positive  side  of  CD  and 
on  the  negative  side  of  AB. 


sin  0     and     cos  (^  zb  ^r) 


cos  0, 


Since 

sin  (^  ih  -)  = 

if  the  signs'  of  all  the  terms  in  (1),  or  (2),  be  changed,  the  direc- 
tion of  AB,  and  also  of  ON,  will  be  changed  by  ±  tt  ;  and  there- 
■  ore  the  positive  and  negative  sides  of  the  line  will  be  reversed. 
That  is,  the  equation  of  a  line  may  be  written  so  as  to  make 
either  side  of  the  line  positive  or  negative,  just  as  we  choose. 

E.  g.    The  equation  of  the  line 
AB, 

(1) 


2 

2 

-2, 

ay  also  be  written 

-I-+ 

\^3y 
2 

-2. 

In(l) 

P 

COSa  = 

=-2, 
=  sin7  = 

1 
=2" 

1 

sin  a=— 

cos  7  = 

— 

t/3 
2 

(2) 


60°    and    7=30= 


♦This  holds  for  all  lines  except  the  x-axls.    (See  §  50.) 


43.] 


THE  STRAIGHT   LINE. 


65 


In  (2)        jp  =  2,    cos  a  =  sin  7 


sin  rt  =  —  cos  y  = 


2  • 


a  =  120°    and    >  =210°. 


Angles  and  directions  corresponding  to  (1)  are  denoted  by  single  arrow- 
heads, those  corresponding  to  (2)  by  double  arrow-heads. 

The  origin  is  on  the  positive  or  negative  side  of  the  line  according  as  the 
equation  is  written  in  the  form  (1)  or  (2). 

Ex.  Point  out  the  combinations  of  signs  of  cos  a,  sin  «,  and  p  when  the 
line  crosses  the  different  quadrants. 

43.      Transformation  of  the  equations  of  the  straight  line. 

In  §§  31,  41,  and  42  we  have  found,  by  independent  methods, 
the  three  standard  forms  of  the  equation  of  a  straight  line  involv- 
ing different  pairs  of  parameters,  m  and  6,  a  and  6,  a  or  y^  andp; 
viz.: 

y  =mx  -\r  b,     Slope  forniy  (1) 


\-^  =  l,     Intercept  form, 


(2) 


{  XC08a-{-ysma=pl  ' 

i       .  '^  >    Distance  or  Normal  form.  (3) 

Ixsmr  —  ycosr=p,  ) 

Any  one  of  these  forms  of  the  equation  may,   however,  be  deduced 
from  any  other. 

I.     From   the  figure  we  obtain 
directly  the  relations 


m  =  tan  y  = 


sm^ 


cos  a 
sin  a 


cosr 

and      p=a  cos  a  =  b  sin  a 

=  —  b  cos  y  =  a  sin  y. 
Then  substituting  these  values  of  m  in  (1),  for  example,  gives 


cos  a 


and 


sin  a 
sin;' 


cosr 


x-^by 
x-\-b. 


56  THE   STRAIGHT   LINE.  [43. 

Whence,  since  h  sin  «  =:=  —  h  cos  y^pj  we  get 

a        0 
X  cos  «  +  ?/  sin  a  =p, 
and  0?  sin  ^  —  y  cos  /  =p. 

Moreover^  the  general  equation  of  the  first  degree ^ 

Ax  +  By  +  C  =  0,  (4) 

can  he  transformed  into  any  one  of  the  three  standard  forms. 

II.  Solving  (4)  for  y  gives  (see  §  31) 

y  =  — -^x  —  -g.      Slope  form,  (5) 

III.  If  we  transpose  and  divide  by  0,  (4)  may  be  written 

• — — J -( ^=1.     Intercept  form.  (6) 

~1       "B 

IV.  To  reduce  the  general  equation  (4)  to  the  distance  form. 

In  this  case  we  are  to  transform  (4)  so  that  the  sum  of  the 
squares  of  the  resulting  coefficients  of  x  and  y  shall  be  unity. 
Hence,  if  we  assume  the  transformed  equation  to  be 

KAx^KBy^KC=0,  (7) 

then  K^A'  +  K'B'  =  cos' «  +  sin' «  =  1 . 

Whence  K  = 


VA'  +  B' 
A  B  C 


•     VA'  +  B'  VA'  -\-B'^  VA'  +  B' 

is  the  required  equation. 

Hence,  to  reduce  the  geheral  equation  (4)  to  the  distance  form,  trans- 
pose C  and  divide  by  V  A^  -\-  B'. 

The  general  equation  of  the  first  degree  must  therefore  repre- 
sent a  straight  line,  since,  by  transposing  and  multiplying  by  a 
suitable  constant,  it  can  be  reduced  to  any  one  of  the  standard 
forms  of  the  equation  of  the  straight  line.     (Cf.  §  31.) 


43.]  THE   STRAIGHT   LINE  57 

V.    Values  of  parameters  in  terms  of  A^  Bj  and  C. 

CompariDg  coefficients  in  (1)  and  (5),  (2)  and  (6),  (3)  and 
(8),  we  get 

C      ^  C  A  —C    ■ 

^^-i'  ^=~B^  ^^-:b'  ^^TI^T^' 

A  .  B 


cos  a  =  sm  r  =     y  ^- ,      sin  a  =  —  COS  r 


\/A'-\-B''  '        VA'-\-B' 

Observe  that  the  values  of  a  and  b  thus  obtained  are  the  same 

as  those  found  by  putting  ?/  =  0,  then  a^  =  0  in  (4);  also  that 

A          b 
m  =  —  ^  = ,  as  found  above  directly  from  the  figure.     Then 

sin  a,  cos  a,  and  p  can  be  found  by  Trigonometry  and  the  relations 
obtained  from  the  figure. 

EXAMPLES. 

1.    When  is  it  impossible  to  write  the  equation  of  a  straight  line  in  the 
intercept  form  f    in  the  slope  form  ? 

Change  the  following  equations  to  the  standard  forms  and  thus  determine 
their  parameters.    Also  draw  the  lines: 


a;  +  ]/%4.lO  =  0.  3.  4^=3x-|-24. 

y^x—6.  5.  5x4-4^  =  20. 

5a;— 12^^13.  7.  ^c  — 4^  +  9  =  0. 

2x  — 3y  =  4.  9.  2x-f3yr=0. 

X  —  a  =  0.  11.  y  =  4. 


4 
6 

8 
10, 

12.  Transform  Ax  -\-  Bi/  -{■  C  =  0  so  that  the  sum  of  the  three  coefficients 
shall  be  K;  so  that  the  square  of  the  first  shall  be  three  times  the  second; 
so  that  the  product  of  the  three  shall  be  twice  their  sum. 

13.  Transform  6x-\-ii/  —  20  =  0  so  that  the  sum  of  the  three  coefficients 
is  22;  so  that  the  product  of  the  first  and  third  shall  be  equal  to  the  second. 

14.  Transform  3ic—  4j/  +  12  =  0  so  that  the  square  of  the  second  coeffi- 
cient shall  be  equal  to  twice  the  third  minus  four  times  the  first;  so  that 
the  product  of  the  three  shall  be  minus  lihree  times  the  last. 

15.  Transform  bx  —  2y  —  3  =  0  so  that  the  product  of  the  first  and  sec- 
ond coefficients  minus  ten  times  the  third  shall  be  equal  to  —  40;  so  that 
the  square  of  the  second  plus  twice  the  sum  of  the  first  and  third  shall  be 
equal  to  24. 


58  THE    STRAIGHT    LINE.  -  [44. 

44.     To  find  the  polar  equation  oj  a  straight  line. 


—X 

Let  iV(p,  a)  be  the  foot  of  the  perpendicular  from  0  upon  the 
given  line  AB. 

Let  P(/>,  0)  be  any  other  point  on  AB. 

Then  lNOP=  (0  —  a), 

and  OP  cos  NOP  =  ON. 

.*.      p  cos  (0  —  a)  =p^  (1) 

which  is  the  required  equation. 

EXAMPLES. 

1 .  Transform  x  cos  a  -|-  ?/  sin  a  =p  to  polar  coordinates. 

2.  What  is  the  polar  equation  of  any  straight  line  through  the  pole? 
of  the  initial  line  ? 

3.  What  locus  is  represented  by  sin  (9  =  0?     sin2(^  =  0?     sin3^=0? 
.  .  .  sinn^  =  0? 

4.  What  is  the  locus  of  cos  n^  =  0  when  n  =  1,  2,  3  .  .  .  ? 
Find  the  parameters  and  draw  the  lines  whose  equations  are 

5.  p  cos  (^  —  30°)  =2.  6.    pGos{e  —  60°)  =  i. 

7.  pcos(^H-45°)=3.  8.    p  cos  (^  +  120°) +  4  =  0. 

9.  pcos(^  — 120°)  +  1  =  0.  10.    p  cos  (^  +  60°) +5  =  0. 

11.  Find  the  coordinates  of  the  point  of  intersection  of  p  cos  (^  ±45°)  =  !. 

12.  Find  the  polar  equations  of  the  bisectors  of  the  angles  between  the 
li^^s  P  cos (6'  — 60°)  =  2,    and    p  cos (^—30°)  =  2. 

13.  What  is  the  polar  equation  of  a  line  perpendicular  to  the  initial 
line  ?    parallel  to  the  initial  line  ? 


4G.J  THE   STRAIGHT   LINE.  59 

14.  Show  that  the  equations 

A  co8d-^B  sin  ^  -f  -  =0,     ^  cot  ^  =  K, 
P==k8ec(0  —  a),  P  =  J  csc(^  — /3), 

represent  straight  lines. 

15.  What  will  the  equation  p  cos(^  —  ^)=P  become  if  the  lines  0  =  a, 
6  =  —  a^e  =  p,d  =  dO°he  taken  as  the  initial  line  ? 

45.  If  we  wish  to  find  the  equation  of  a  straight  line  which 
satisfies  any  two  conditions,  we  may  take  for  its  equation  any 
one  of  the  forms 

X       y 
y  =  mx-\-b,     -  +  |-  =  1,     lx-i-mij  =  l, 

X  cos  a  -f-  ^  sin  a  =  p,     Ax  -f-  Bij  +  0=0. 

The  given  conditions  must  be  expressed  by  two  equations  in- 
volving the  parameters  of  the  line.  From  these  tim  equations  of 
condition  we  have  then  to  determine  the  values  of  one  of  the 

©airs  of  parameters, 

A  B 

m  and  b,     a  and  b,     I  and  m,     p  and  a.     or     — ^  and  -^. 

If  the  line  be  made  to  satisfy  only  one  condition,  there  will  be 
only  one  conditional  equation  involving  the  parameters,  and  con- 
sequently only  one  parameter  can  be  eliminated ;  then  by  assign- 
ing a  suitable  value  to  the  remaining  parameter,  the  line  may  be 
made  to  satisfy  any  other  given  condition. 

Ex.  Find  the  equation  of  a  straight  line  passing  through  the  point 
( — 4,  1)  and  making  equal  intercepts  on  the  axes. 

Let  — + 1-  =  1  be  the  equation  of  the  line. 

Then,  since  the  intercepts  are  equal,  a^b. 

1                                —4      1 
Also,  since  ( — 4,  1)  is  on  the  line, f-  ^  =  1. 

. • .    a  =  b  = — 3,    and    x-\-y-\-S==0    is  the  equation  required. 

46.  To  find  the  equation  of  a  straight  line  passing  through  a  fixed 
point  (o^i,  i/i)  in  a  given  direction. 

Let  the  line  make  with  the  a:-axis  an  angle  tan~*m. 
Its  equation  will  then  be  (where  b  is  unknown) 

y=mx  +  b;  (1) 

and  since  the  line  passes  through  (x^,  ^/i), 

2/i  =  mj?i  +  6.  (2) 


60  THE   STRAIGHT   LINE.  [47. 

Whence,  by  subtracting  (2)  from  (1), 

y  —  y^  =  m(x  —  Xi).  (3) 

The  line  given  by  (3)  will  pass  through  the  point  (x^  y^)  for 
all  values  of  m ;  and  may  be  made  to  represent  any  line  through. 
{Xi,  yi)  by  giving  to  m  a  suitable  value. 

If  then  we  know  a  line  passes  through  a  certain  point  we  may 
write  its  equation  in  the  form  (3),  and  determine  the  value  of  m 
from  the  other  condition  the  line  is  made  to  satisfy. 

sin  Y 
Since  m=  tan  r= (§  42),  equation  (3)  may  be  written 

in  the  form 

^ZZj^^lSZJb^r,  (4) 

COS  r  sin  y 

where  r  is  the  variable  distance  from  the  fixed  point  (Xi,  y^)  to 
any  point  (x,  y)  on  the  line. 

Let  the  student  prove  (4)  directly  from  a  figure. 

47.  To  find  the  equation  of  a  straight  line  which  passes  through  two 
given  points  {x^,  y^)  and  {x^,  y^)- 

Since  the  line  passes  through  (xi,  y^)  its  equation  will  be  of 
the  form  [(3),  §  46] 

y  —  y,  =  m(x  —  Xi)',  (1) 

then,  since  (x2,  2/2)  is  also  on  the  line,  we  have 

^2  — 2/i  =  *^(^2  — ^1).  (2) 

Dividing  (1)  by  (2)  gives  the  required  equation 
y  —  yi^x  —  x, 

2/2  —  2/1       0C2  —  Xi 
Equation  (3)  may  also  be  written 

X,   y,    1 

oci,  2/1,  1=0,  (4) 

^2  J  2/2  J  ■'- 

which  is  obvious,  since  the  area  of  the  triangle  formed  by  (a^j,  2/1) 
(X2,  2/2)  ^^^  ^^J  other  point  (x,  y)  on  the  line  is  zero. 


47.]  THE   STRAIGHT   LINE.  .   61 

EXAMPLES. 
Find  the  equation  of  the  straight  line 

I.  if  6  =  §  and  >  =tan-' ^.  2.    if  a  =  5  and  p  =5. 

'  3.    if  7  =  30°  and  i> =4.  4.    if  6 =—3  and  7  =  150°. 

6.    if  7  =  tan-i  2  and  the  line  passes  through  (3,  —  4). 

6.  if  7  =  tan"^  j-  and  the  line  passes  through  ( —  a,  6). 

7.  passing  through  the  pairs  of  points  (2,  3)  and  ( — 6,  1); 
(—  1,  3)  and  (6,  —  7) ;    (a,  b)  and  (a  +  6,  a  —  5). 

8.  Find  the  equations  of  the  sides  of  the  triangle  whose  vertices  are 
the  points  (1,  3),  (3,  —5)  and  (—1,  —3). 

9.  Find  the  equations  of  the  three  medians  of  this  triangle,  and  show 
that  they  meet  in  a  point. 

10.  Find  the  equation  of  a  line  passing  through  (—  1,  4)  and  having  in- 
tercepts (1)  equal  in  length,  (2)  equal  in  length  but  opposite  in  sign. 

II.  Show  that  the  equations  of  the  lines  passing  through  the  point  (4, 4) 
and  whose  distance  from  the  origin  is  2  are  x(l  zh  i/7)  +  y{i  =F  i/l)=8. 

12.  Find  the  equation  of  the  line  through  (a,  b)  parallel  to  the  line  join- 
ing (0,— a)  and  (6,0). 

13.  What  is  the  equation  of  the  line  through  (4, — 5)  parallel  to 
2a;  +  3i/  =  6? 

14.  Find  the  equations  of  the  lines  which  pass  through  ( — 2,  1)  and  cut 
off  equal  lengths  from  the  axes. 

1^.    Show  that  the  three  lines  2ac  —  y  =  4,    x+2y  =  7,    and    3a;  -f-  y  =  1 1 
t  in  a  point. 

16.  Show  that  the  three  points  (1,  3),  (—1,  4),  and  (9,  —1)  are  on  a 
straight  line;  also  (3a,  0),  (0,  3b),  and  (a,  26). 

17.  Show  that  the  equation  of  the  line  passing  through  the  points 
{a  cos  a,  6  sin  a)  and  (a  cos  /?,  6  sin  p)  is 

6a;  cos  iia  +  l3)-]-ay  sin  ^(a  +  /3)  =  a6  cos  i{a  —  /3). 

18.  Show  that  the  equation  of  the  line  which  passes  through  the  points 
<a  sec  a,  6  tan  a)  and  (a  sec  /?,  6  tan  /?)  is 

6a;  cos  -^(a  —  /?)  —  ay  sin  -J(a  -j- /?)  =a6  cos  ^a  -f  j3). 

19.  Find  the  equations  of  the  lines  which  bisect  the  opposite  sides  of 
the  quadrilateral  (3,  4),  (5,  1),  (—3,  4),  and  (5,-1). 

20.  Find  the  equations  of  the  lines  which  go  through  the  origin  and 
trisect  that  portion  of  the  line  3a; — 2y  =  18  which  is  intercepted  between 
the  axes. 


62 


THE    STRAIGHT    IJNE. 


[48. 


21.  For  what  value  of  m  will  the  line  y^mx  —4  pass  through  (4,  2)? 
be  2  units  distant  from  the  origin  ? 

22.^  A  line  is  3  units  distant  from  O  and  makes  an  angle  of  60°  with  OX. 
What  is  its  polar  equation  ?    its  rectangular  equation  ? 

23.  Find  the  locus  of  all  points  which  are  equidistant  from  the  two  lines 
3a;— 2^  =  8    and    3x— 2?/-f2  =  0. 

24.  What  is  the  distance  between  the  parallel  lines 

3x4-4^  =  5    and    6x-f 8^-{-15  =  0? 

25.  Show,  by  the  use  of  (1),  §  44,  or  by  transforming  (3),  §  46,  that  the 
polar  equation  of  a  line  passing  through  the  fixed  point  (|0i,  ^i)  may  be 
written 

p  COS(^  —  n)  =  p^  C0S(^i  —  a). 

26.  Show,  directly  from  a  figure,  or  by  transforming  (3),  §  47,  that  the 
polar  equation  of  the  straight  line  which  passes  through  the  two  fixed 
points  (pi,  ^i)  and  (/a2,  ^2)  is 

P1P2  sin  {O2 — ^1)  +  P2P  sin  (^  —  ^2)  +  {PP\  sin  (^1  —  ^)  =  0. 

27.  Show  that  the  three  straight  lines 

ttix  +  hxy  +  Ci  =  0,     aix  -\-  h^y  +  Ca  =  0,     a^x  -{-h-^y -\-Cz=0 
will  meet  in  a  point  if 

ai,  5i,  Ci 

a2 ,  62  >  C2   =  0. 
as,   &3,  Cs 

28.  Find  the  determinant  expressions  for  the  coordinates  of  the  vertices, 
and  for  the  area  of  the  triangle  formed  by  the  three  lines  in  Ex.  27,  and 
show  that  the  determinant  there  given  is  a  square  factor  of  the  determi- 
nant expression  for  the  area  of  the  triangle. 


48.      To  find  the  angle  between  two  straight  lines  whose  equations  are 
given. 


Let  AB  and  A'B'  be  the  given  lines. 


48.]  THE    STRAIGHT    LINE.  63 

Let  ip  be  the  required  angle. 
'  Then,  using  the  same  notation  and  the  same  convention  as  to 
direction  of  the  lines'  as  in  §  42, 

^p  —  a  —  a'—y—y'.  (l) 

I.  If  the  equations  of  the  given  lines  be 

X  cos  a  -|-  7/  sin  a  =  'p-    and     x  cos  o-'  -\-  y  sin  a'  —  jo', 
cos  tp  can  be  found  by  direct  substitution  in 

cos  fp  =  cos  a  COS  a'  -f"  sin  a  sin  a'.  (2) 

II.  If  the  equations  of  the  given  lines  be 

y  =  mx  -\-  b     and     y  =  m'x  +  ^'j 
we  have  from  (1),  since  tan  y  —  m,  and  tan  /  =  m', 

.  ,x  tan  7-  —  i^nf        m  —  m' 

tan  ^  =  tan  (r  -  r')  =  i  +  tan  ,  tan"?  =  1+^-        ^^) 

...     ,  =  tan-(i?L=^).  (4) 

When  m  =  m',  tan  ^  =  0,  and  the  lines  are  parallel. 
When  1  -|-  mm!  —  0,  tan  (p  is  infinite. 

Therefore,  when  m'  = the  lines  are  perpendicular  to  one 

m 

another. 

III.  If  the  equations  of  the  lines  be 

^^  +  %  +  C  =  0     and     A'x  +  B'y  +  C  =  0, 

A  A' 

then     m  =  — ^,      m'  —  —  -^^;     and  therefore,  from  (3), 

X)  X) 


A'B  —  AB'  ,_, 

If  A'B  —  AB'  —  0,  i.  e.  if  -77  =  -d7,  the  lines  will  be  parallel. 

If  AA'-\-  BB'=  0,  the  lines  will  be  at  right  angles  to  one  an- 
other. 

It  should  be  noticed  that  (2)  gives  the  angle  between  two  di- 
rected lines.  For  if  all  the  signs  in  one  of  the  equations  in  I.  be 
changed,  the  direction  of  the  line  will  be  changed  by  ±:  tt,  (§  42). 


64  THE   STRAIGHT   LINE.  [49. 

The  sign  of  cos  y  given  by  (2)  will  also  be  changed  and  <p  be- 
comes the  supplement  of  its  former  value.  But  if  all  the  signs 
in  both  equations  be  changed,  ^  is  unaltered. 

If  the  equations  be  so  written  that  the  origin  is  on  the  same 
side  (either  positive  or  negative)  of  both  lineS;  it  will  be  in  the 
obtuse  angle  between  the  lines  when  cos  <p  is  positive,  and  in  the 
acute  angle  when  cos  <p  is  negative. 

If  m  and  m'  be  so  taken  that  m'  >  m,  then  /  >  r  and  (3)  will 
give  tan  ( —  ^)  =  — tan  ^,  instead  of  tan  (p. 

49.     To  find  the  equations  of  two  lines  passing  through  a  given  point 
(^i>  2/i)j  ^^^  one  parallel,  the  other  perpendicular  to  a  given  line. 
Let  the  given  line  be 

Then  the  parallel  line  is 

Ax  +  By^K=(),  [§48,  IIL]      (1) 

and  the  perpendicular  line  is 

Bx—Ay+K'^0,  [§48,  III.]      (2) 

where  ^and  K'  are  constants  to  be  determined. 

Since  both  (1)  and  (2)  are  to  go  through  (a^j,  t/j  these  con- 
stants are  such  that 


Ax,  -i-By.-^K 
and  Bx, 


(3) 


i.e.  K=-(Ax,  +  By.)    \ 

and  K'  =  —  {Bx,  —  Aij,).  J  ^^ 

Therefore,  the  required  equations  are,  respectively, 

A{x-x,)-\-B{y  —  y,)^(}  (5) 

and  B(ix  —  x,)  —  A{y  —  y,)=0.  (6) 

If  the  equation  of  the  given  line  is  in  the  form 

y  =  mx  +  h, 

the  required  equations  may  be  written  [(3),  §  46,  and  II. ,  §  48] 

y  —  yi  =  m(x  —  x,)  (7) 

and  y  —  yi  =  —-(x  —  3Ci)-  (8) 


49.]  THE   STRAIGHT   LINE.  65 

EXAMPLES. 
Find  the  angles  between  the  following  pairs  of  lines: 
(p3xH-4y=8    and    ly  —  x-\-U  =  0. 

2.  2x-f-3i/=6    and    2y  =  3x  —  12. 

3.  x-\-i=2y    and    x  +  Sy  =9. 

4.  3i/  +  12x 4-16  =  0    and    2y=ix-\-bo 

,f5:^^y=i    and     ^-^-  =  1. 
(J/  a       b  a      0 

d?  Prove  that  the  points  (1,  3),  (5,  0),  (0,-4),  and  (—4,-1)  are  the 
vertices  of  a  parallelogram,  and  find  the  angle  between  its  diagonals. 

Find  the  equations  of  the  two  straight  lines 

nJ  passing  through  the  point  (2,  3),  the  one  parallel,  the  other  perpen- 
dicular to  the  line  ix  —  3y  =  6. 

j8.y  passing  through  (4,  —  2),  the  one  parallel,  the  other  perpendicular  to 
thWline  y  =  2x  +  4. 

/9/  passing  through  the  intersection  of  4x  -f  2/  +  5  =  0  and 
2ic  —  3i/+  13  =  0,  one  parallel,  the  other  perpendicular  to  the  line  through 
the  two  points  (3,  1)  and  ( —  1,  —  2). 

(lOJ  Find  the  equation  of  the  perpendicular  bisector  of  the  line  joining 
thfepoints  (3,  —  1)  and  (—  2,  1).  * 

11.  Find  the  equations  of  the  lines  perpendicular  to  the  line  Joining 
(2,  1)  and  ( —  3,  —  2)  at  the  points  which  divide  it  internally  and  externally 
in  the  ratio  2:3. 

12.  What  is  the  equation  of  a  line  parallel  to  Sx-\-4:y  =  12  and  at  a  dis- 
tance 4  from  the  origin? 

13.  Show  that  two  parallel  lines  intersect  at  infinity. 
The  vertices  of  a  triangle  are  (3,  1),  (—  2,  3),  and  (2,-4): 

iijl.  Find  the  equations  of  its  altitudes  and  show  that  they  meet  in  a 
point. 

15.  Find  the  equations  of  the  perpendicular  bisectors  of  its  sides,  and 
show  that  they  meet  in  a  point  which  is  equidistant  from  the  three  vertices. 

,6.    Find  its  interior  angles. 


d 


J/^    Find  the  equations  of  two  lines  through  the  origin,  each  making  an 
angle  of  30°  with  the  line  4x  +  2/  +  4  =  0. 

18.    Show  that  the  equations  of  the  two  straight  lines  through  a  given 
point  [xi ,  2/1)  making  a  given  angle  ^  with  the  line  y  =mx-\-b  are 


mzfctan^  , 
^      ^       lq=mtan^^         ^^ 


6 


66 


THE    STKAIGHT   LINE. 


[50. 


19.  Show  that  the  equations  of  the  lines  passing  through  ( —  3,  2)  and 
inclined  at  an  angle  of  60°  to  the  line  V %  —  x  =  3  are 

a;4-3  =  0    and     l%  +  x  +  3  =  0. 

20.  Find  the  equations  of  the  sides  of  a  square  of  which  the  points  (2, 2) 
and  ( —  2,  1)  are  opposite  vertices. 

21.  What  are  the  equations  of  the  sides  of  a  rhombus  if  two  opposite 
vertices  are  at  the  points  ( —  1,  3)  and  (5,  —  3),  and  the  interior  angles  at 
these  vertices  are  each  60°  ? 

22.  Prove  that  the  equation  of  the  straight  line  which  passes  through 
the  point  (a  cos^  ^,  a  sin^  6)  and  is  perpendicular  to  the  straight  line 
X  sec  0-\-y  CSC  6^  =  a  is 

X  cos  0  —  y  sinO  =  a  cos  2d. 

50.  To  find  the  perpendicular  distance  from  a  given  straight  line  to 
a  given  point  Fi(xi,  yi). 


Let  HKhe  the  given  line,  and  let  H'K'  be  parallel  to  HK  a,nd 
pass  through  Pj. 

Let  Pi  Q  be  the  perpendicular  from  Pj  on  HK,  and  OR,  OB'  the 
perpendiculars  from  0  on  UK  and  H'K, 

Let  the  equation  of  HK  be 

X  cos  a  -\-  y  Qin  a  =  p.  (1) 

Then  the  equation  of  HK  is 

X  cos  a-{-  y  sin  a=p  -{- BE'  =^p  +  QP^ ;  (2) 

and  since  this  line  (2)  goes  through  Pi(a^i,  y^), 

Xi  cos  «  +  2/i  sill  «  =P  +  Q^i'  (3) 


50.]  THE  STRAIGHT   LINE.  67 

.♦.     §Pj  =  Xi  COS  «  +  2/1  ^i^  "  — i^>*  (^) 

which  is  the  distance /rom  the  line  «,  ^  to  the  point  (aJi,  t/i). 
If  the  equation  of  the  given  line  be 

Ax-\-By-{-C  =  0, 
A  .  B 

cos  a  =  —-=====-,      sin  a  = 


VA^^-B''  VA'  +  B" 

and  substituting  these  values  in  (4)  gives 

Axi  -\-By,-\-G 

which  is  the  distance  from  line  A,  B,  C  to  the  point  (a?i,  2/1  )• 

Hence  the  length  of  the  perpendicular  from  a  given  line  to  a  given 
point  is  found  by  substituting  the  coordinates  of  the  point  in  the  equation 
of  the  line  reduced  to  the  distance  form  with  all  the  terms  transposed  to 
the  first  member. 

The  expression  (5)  will  be  positive  or  negative  according  as 
Axi  +  By  I  +  018  positive  or  negative  (if  VA^  +  B^  be  positive). 
If  Axi  +  By^  +  Ois  positive,  the  point  {x^,  ^/l)  is  said  to  be  on 
the  positive  side  of  the  line  Ax  +  By  -j-  0  ==  0 ;  if  Ax^^  -\-  By^  +  C 
is  negative,  (a?i,  i/O  is  said  to  be  on  the  negative  side  of  the  line. 
If  the  equation  of  the  line  be  written  so  that  p  is  positive,  the  ex- 
pression (5)  will  be  found  to  be  positive  when  Pj  and  0  are  on 
opposite  sides  of  the  line.     (  Of.  §  42. ) 

Hence  the  points  {xi,  y^)  and  {x^,  2/2)  are  on  the  same  side  or 
opposite  sides  of  the  line  Ax  -\-  By  -\-  C  =  0  according  as 
Axi  -j-  By^  +  C  and  Ax.^  +  By^  -\-  C  have  the  same  sign  or  opposite 
signs. 

This  proves  for  the  straight  line  the  principles  illustrated  in 
§§  14-20. 

*  Another  proof ,  Let  the  coordinates  of  Q  be  Xo,  y^;  of  D,  xi,  2/2;  then 
«2  COS  a  +  2/2  sin  a  =Pi  since  Q  is  on  (1).    Projecting  on  QPy  gives 

QPi  =  proj.  QD  +  proj.  DP^ 

.  •.     QPi  =  (xi  —  X2)  cos  a  +  (2/1  —  2/2)  sin  a 

—  {Xi  cos  a  +  2/1  sin  a)  —  (a^  cos  a  +  2/2  sin  a) 

=  Xx  COS  a  +  2/1  sin  a  — p. 


68  THE   STRAIGHT   LINE.  [51. 

51.  To  find  the  equations  of  the  bisectors  of  the  angles  between  the 
lines 

Ax  -f  By  -|-  C  =  0,     or     x  cos  «  +  ?/  sin  «  — p  —  0,      (1) 

and     A'x  -|-  B'y  -f  0'=  0,     or     a?  cos  a'-j-  y  sin  «' — 'p'  —  0.      (2) 

Suppose  the  equations  of  the  lines  written  so  that  the  origin  is 
on  the  same  side  of  both  lines. 

Then  for  any  point  (x,  y^  on  the  bisector  of  the  angle  which 
includes  the  origin, 

Dist.  from  (1)  to  (x,  y)  ==  Dist.  from  (2)  to  (x,  y) ; 

and  for  any  point  (x,  y)  on  the  other  bisector, 

Dist.  from  (1)  to  (x,  y)  =  — Dist.  from  {2)  to  {x,  y). 

Therefore  the  required  equations  are  [§  50] 

Ax-^By+G  ^  ^  A'x -\- B'y-]- a 
■VA'-\-B'         -      VA"-\-B''    '  ^   ^ 

or  X  cos  a-  -\-  y  sin  a  — p  —  ±  x  cos  a'-\-  y  sin  «' — p'.        (4) 

Ex.    Show  that  these  two  lines  are  perpendicular  to  each  other.    [Use  (4).] 

EXAMPLES. 
Find  the  following  distances : 

1.  From  3x4-42/ +  10  =  0    to    (1,12),     (-3,-9),    (3,4). 

2.  From  a;  — 32/ =  7    to    (3,2),    (6,3),     (2,-5). 

3.  From  5^+122/ =  13    to    (3,-2),    (—3,2),    (4,-7). 

4.  From  5(a;  — a)+a(2/  — 6)  =0    to    (—a, —6),    (— &,  — a),    (6,  a). 

5.  From4(a;-3)  =  3(2/  +  l)    to    (6,1),    (4,-5),    (-7,2). 
Are  the  above  points  on  the  same  or  opposite  sides  of  the  lines  ? 
Find  the  equations  of  the  bisectors  of  the  angles  between  the  lines 

6.  3x  +  42/+12  =  0    and    4x  — 32/  =  12. 

7.  3x  — 42/  +  5  =  0    and     12x  +  51/ + 14  =  0. 

8.  2/  =  2x+5    and    x  —  2y  =  S. 

9.  y=  VSx  +  3    and    x+  VSy  =  9, 

10.  Find  the  lengths  of  the  altitudes  of  the  triangle  whose  vertices  are 

(3,4),  (-4,1),  and  (-1,-5). 

11.  What  is  the  locus  of  a  point  which  is  3  units  distant  from  the  line 
2a;  — 42/ =  99 


52.]      ,  THE   STRAIGHT   LINE.  69 

12.  Find  the  points  on  the  axes  which  are  4  units  from  the  line 
x-ly-\-21  =  0. 

13.  Show  that  the  perpendiculars  let  fall  from  any  point  of  22a; — iy  =  15 
upon  the  lines  24a;  -\-ly  =  20  and  4a; — Sy  =  2  are  equal.  Find  another  line 
of  which  this  statement  is  true. 

14.  Find  the  perpendicular  distance  of  the  point  (I,  m)  from  the  line 
through  (a,  6)  perpendicular  to  Ix  +  wi/  =  1. 

15.  Show  that  the  bisectors  of  the  interior  angles  of  a  triangle  meet  in 
a  point. 

16.  Find  the  locus  of  a  point  which  is  equally  distant  from  the  lines 
5a;  — 32^  =  15 and  Sy  =  5x-\-6. 

17.  Show  that  the  locus  of  a  point  which  moves  so  that  the  sum  of  its 
distances  from  the  two  lines 

X  cos  «  +  2/  sin  a=p    and    x  cos  a^  -\-y  sin  a^  =  p^ 
is  constant  and  equal  to  ^  is  the  straight  line 

X  cos  K«  +  «^)  +  2/  sin  ^(a  -f  a^)  =  2{p  -{-p^  -^K)  sec  ^(a  -|-  a^). 

Show  that  the  locus  is  parallel  to  one  of  the  bisectors  of  the  angles 
formed  by  the  two  given  lines. 

Show  also  that  if  the  difference  of  the  distances  from  the  two  given  lines 
is  constant,  the  locus  is  a  straight  line  parallel  to  the  other  bisector. 

18.  If  p  and  p^  be  the  perpendiculars  from  the  origin  upon  the  straight 
lines  whose  equations  are 

X  sec  ^  +  2/  CSC  0  =  a    and    x  cos  0  —  y  sind  =  a  cos  2^, 

prove  that  Ap^-\-p^^  =  a'^. 

52.  To  find  the  equation  of  a  straight  line  passing  through  the  irir 
tersection  of  two  given  straight  lines. 

The  most  obvious  method  of  finding  the  required  equation  is 
to  find  the  coordinates  x'y  y'  of  the  point  of  intersection  of  the 
two  given  lines,  and  then  substitute  these  values  in  equation  (3), 
§46.    • 

The  following  method  of  dealing  with  this  class  of  problems 
is,  however,  sometimes  preferable,  both  on  account  of  its  gener- 
ality and  because  it  saves  the  labor  of  solving  the  two  given 
equations : 

Let  the  equations  of  the  two  given  straight  lines  be 

Ax-^By+C  =  0,  (1) 

and  A'x  +  B'y  -|-  C'=  0.  .  (2) 


70  THE   STRAIGHT   LINE.  [52. 

The  required  equation  is  then  written 

Ax -i- By -\- C -\-  X(A'x  +  B'y  +  C")  =  0,  (3) 

where  ^  is  any  constant. 

Equation  (3)  is  of  the  first  degree,  and  therefore  represents  a 
straight  line;  if  (a?',  y')  is  the  point  common  to  (1)  and  (2),  we 
have 

Ax'+By'-^  C=0 

and  ^V+^Y+C"=0. 

.-.     Ax'-^Bx'-^C-}-K^'x'-\-B'y'-\-a)=0, 

which  shows  that  the  point  (x',  y')  is  also  on  (3). 

Hence  (3)  is  the  equation  of  a  straigJit  line  passing  through 
the  point  of  intersection  of  the  two  given  lines.  Moreover, 
equation  (3)  contains  one  arbitrary  parameter,  A,  and  therefore, 
by  giving  a  suitable  value  to  A,  the  line  may  be  made  to  satisfy 
any  other  given  condition ;  it  may,  for  example,  be  made  to  pass 
through  any  other  given  point,  may  be  made  parallel,  or  per- 
pendicular to  a  given  line.  Hence  equation  (3)  represents,  for 
different  values  of  A,  all  straight  lines  through  the  point  of  in- 
tersection of  (1)  and  (2). 

The  other  condition  which  any  particular  line  is  made  to  sat- 
isfy will  give  an  equation  for  the  determination  of  the  value  of  A. 

Ex.    Find  the  equation  of  a  straight  line  passing  through  the  point  of 

intersection  of  2a;  -f  63/  —  4  =  0  and  4a;  —  22/  +  2  =  0,  and  perpendicular  to 

the  line 

2x  —  iy  =  l.  (1) 

Any  line  through  the  intersection  is  given  by 

2x  +  52/  —  4  +  A(4a;  —  22/  +  2)  =  0, 

or  (2  +  4/0a;  +  (5  — 2;i)2/  +  (2A-4)  =  0.  (2) 

Now  (2)  is  perpendicular  to  (1)  if  (§  48,  III.) 

2(2  +  4;^)  — 4(5  — 2a)  =  0;    -i.  e.,  if  A  =  1. 

.-.    6a;  +  31/  =  2  is  the  required  equation. 

EXAMPLES. 

1.  Find  the  equations  of  the  lines  joining  the  points  (0,  0),  (4,  2), 
( —  1,  3),  ( — 3,  —  4)  to  the  point  of  intersection  of  the  lines  2x-\-y  =  2  and 
2x  —  Sy  =  6. 

2.  What  is  the  equation  of  the  straight  line  passing  through  the  inter- 
section of  4a;  —  22/  =  4  and  lx  —  3y-{-2i=  0,  and  parallel  to  9a;  —  42/  =  0 ? 


52.]  THE  STRAIGHT  LINE.  71 

3.  Find  the  equations  of  the  two  lines  passing  through  the  intersection 
oix  —  2y  =  i  and  2x  +  %  +  4  =  0,  the  one  parallel,  the  other  perpendicu- 
lar to  a;  +  2y  =  0. 

4.  Find  the  equations  of  the  two  lines  passing  through  the  intersection 
of  7x  —  52/  =  35  and  8x  —  3?/  +  24  =  0,  the  one  parallel  to  i/  =  2x,  the  other 
perpendicular  to  3a;  -f  42/  =  0. 

5.  What  is  the  equation  of  a  line  passing  through  the  intersection  of 
3a;  —  22/  +  12  =  0  and  x-{-iy  =  20,  and  (a)  equally  inclined  to  the  axes? 
(b)  whose  slope  is  —  2? 

6.  The  distance  of  a  line  from  the  origin  is  5,  and  it  passes  through  the 
intersection  oi  2x -\- 3y -{- 11  =  0  and  3a;  —  by  =  16.    Find  its  equation. 

7.  Find  the  equations  of  the  two  lines  which  pass  through  the  intersec- 
tion otx-\-2y  =  0  and  2a; — 2/  +  8  =  0,  and  touch  the  circle 

a;2  +  2/'  =  9- 

8.  Find  the  equations  of  the  two  lines  which  pass  through  the  intersec- 
tion ofx-\-3y-\-9  =  0  and  Sx  =  y  -\- 13,  and  touch  the  circle 

ix  +  2y  +  (iy-3y  =  25. 

9.  Find  the  equations  of  the  diagonals  of  the  rectangle  whose  sides  are 
a;  +  22/  =  10,  a;  +  22/  +  2  =  0,  2a;  —  2/  =  12,  and  2a;  —  2/  =  16,  without  finding 
the  coordinates  of  its  vertices. 

10.  Show  that  it  S  =  0  and  5^^=  0  represent  the  equations  of  any  two  loci 
with  terms  all  transposed  to  the  first  member,  and  A  denotes  an  arbitrary 
constant,  then  the  locus  represented  by  the  equation 

will  pass  through  all  the  common  points  of  the  two  giyen  loci. 
Consider  the  two  cases  A  =  0,  and  A  =  00 . 

11.  Find  the  equation  of  the  circle  which  passes  through  the  origin  and 
the  common  points  of  the  circles 

a;2  +  ^2^25    and    x^ -{- y^  —  ISx  +  20  =  0. 

12.  Find  the  equation  of  the  circle  which  passes  through  the  common 
points  of 

x2  -f  2/^  =  16    and    x  —  2/  =  4, 

and  (1)  passes  through  the  origin,  (2)  touches  the  a;-axis. 

13.  A  circle  passes  through  the  common  points  of 

a;2_|_2/2^26    and    a;  — 42/  + 13  =  0, 
and  cuts  the  a;-axis  in  two  coincident  points.    Find  its  equation. 


72  THE   STRAIGHT   LINE.  [53. 


Equations  Representing  Two  or  More  Straight  Lines. 

53.*     The  straight  lines  represented  by  n  equations  of  the  first  degree 
may  be  represented  by  a  single  equation  of  the  nth  degree. 

Let  S;  =  AiX-\-B,y  -^  C,=0, 

S,  =  A,X^B,y-\-C,=  0, 


S„  =  A.X^B„y-^C^  =  0, 

be  the  equations  of  n  straight  lines. 

Taking  the  product  of  these  n  expressions  gives 

S,SA  .  .   .  /S„  =  0.  (1) 

Equation  (1)  is  satisfied  by  the  coordinates  of  all  the  points,  and 
no  others,  which  satisfy  the  separate  equations  Si—0,...S,^=0; 
because  the  product  S-^SA  •  .  .  /S'„  =  0  when,  and  only  when,  at 
least  one  of  its  factors  is  zero.  Therefore  all  points  which  are  on 
the  n  given  lines,  and  no  other  points,  are  on  the  locus  of  (1). 
But  (1)  is  an  equation  of  the  nth  degree,  hence  the  proposition. 

Conversely,  if  an  expression  of  the  nth  degree  can  be  separated  into  n 
factors  of  the  first  degree,  it  will  represent  n  straight  lines^  when  equated 
to  zero. 

Observe  that  in  the  given  equations  all  terms  must  be  trans- 
posed to  the  first  member  before  we  multiply  or  factor. 

E.  g.    Let  the  given  equations  hey  —  x  —  a  =  0,  and  y  -[-x  —  a  =  0. 
Since  {y  —  x~a){y-\-x  —  a)=y'^  —  x'^  —  2ay-\-a'^,  (1) 

the  same  locus  will  be  represented  by 

(2/  — x  — a)(i/  +  x  — a)  =  0    and    y''  —  x^  —  2ay-^a'  =  Q.  (2) 

The  given  equations  may,  however,  be  written 

■  3)    f)  =  a  +  x,    or    (4)    y~x  =  a, 
and  (5)    y  =  a  —  x,    or    (6)    y-\-x  =  a. 

Multiplying  (3)  by  (5)  and  (4)  by  (6)  gives,  respectively, 
(7)    y'^  =  a^  —  x^,    or    x"^  -\-y^  =  a'^j    a  circle; 
and         (8)    y^  —  x'^  =  a%    a  hyperbola  (§  36),  instead  of  two  lines. 

The  first  members  of  (2)  are  identities,  i,  e.,  the  same  for  all  values  of  x 
and  y;  but  equations  (3),  (5),  (7),  and  (8)  are  merely  consistent;  i.  e.,  they 
*  For  this  reason  factors  of  the  first  degree  are  sometimes  called  linear  factors. 


64.]  THE   STRAIGHT   LINE.  73 

are  satisfied  by  the  pair  of  values  x  =  0,  2/  =  «>  and  by  no  other  real  pair. 
Hence  we  observe  that  the  loci  of  identities  equated  to  zero  coincide, 
whereas  the  loci  of  consistent  equations  merely  concur. 

Ex.  Show  that  the  equation  SiS-^S^  .  .  .  Sn  =  0  represents  all  the  loci  of 
Si=Of  &2  =  0,  83  =  0  .  .  .  <Sn  =  0,  whatever  the  form  of  the  expressions 
Si,  S.>,  Sz  .  .  .  Sn  may  be. 

.    Is  the  locus  of  ~  =0,  or  Si -{-82  =  0,  the  same  as  the  loci  of  fii  =  0  and 

The  equation  of  any  two  straight  lines  may  therefore  be  writ- 
ten in  the  form 

(Ix  -\-my  -\-n)  (Vx  +  m'y  +  n')  =  0, 
or  IVaf  +  (^wi'  +  l'ni)xy  +  mm'y^  +  (W  +  rn)x 

-f-  (mw'  4"  in'n^y  -f  nn'=  0. 

This  equation  contains  terms  involving  ar*,  y^,  xy,  x,  y,  and  a 
constant,  or  all  possible  terms  involving  x  and  y  of  a,  degree  not 
higher  than  the  second.  A  notation  which  is  in  general  use  for 
such  an  expression  is 

ax'  +  2hxy  +  h/  +  2gx  +  2fy  +  c, 

and  when  equated  to  zero  is  called  The  General  Equation  of 
the  Second  Degree. 

Hence,  the  most  general  equation  which  represents  two  straight 
lines  is  a  form  assumed  by  the  general  equation  of  the  second 
degree.  An  equation  of  the  second  degree,  however,  can  not  be 
separated  into  linear  factors  unless  a  certain  relation  holds  be- 
tween the  coefficients  of  its  terms,  and  therefore  does  not  always 
represent  a  line  pair.  E.  g.,  x'±  y^=l,  y'=  x,  xy  =  a'  are  not 
line  pairs. 

In  general,  as  will  be  shown  in  Chap.  VII,  an  equation  of  the 
second  degree  represents  a  Conic  Section. 

54.*  To  find  the  condition  that  the  general  equation  of  the  second 
degree  may  represent  two  straight  lines. 

The  necessary  and  sufficient  condition  that  (§  53) 
ax"  +  2hxy  +  hif  ^2gx -{■2jy  ^  c  =  0 
may  represent  a  pair  of  straight  lines  is 
ax'  +  "^h^y  +  hy'  +  2gx  +  2/?/  +  c 

=  (Ix  -\- my  -\- n)  (Vx  +  m'y  -f  %'), 


74  THE   STRAIGHT   LINE.  [55. 

These  two  expressions  are  identically  equal  if  their  coefficients 
are  respectively  equal;  i.  e.,  if  (§  53) 

a  =  W,     b  =  mm',     c  =  nn', 

2h  =  Im'  +  Vm,     2g  =  In'  +  Un,     2/  =  mn'  -f  m'n. 

The  continued  multiplication  of  the  last  three  of  these  equations 
gives 
Sfgh  =  2lUmm'nn'-\-  ll\m^n'^  +  m'V)+  mm'^nH'''  +  n'T) 

H-nri'C^V^  +  Z'W) 

=  2lVmm'nn'-\-  lV\^(jnn'-\-  m'ny  —  2mm'nn'~\ 

+mm'[(7ir+  n'iy  —  2nn'lV']  +  nn'[(lm'^  Vmy  —  2lVmm''] 

=  2abc  +  a(4/  —  26c)  +  6(4^^  — 2ac)  +  c(4/i2  —  2a6). 

.-.     a6c  +  2/^/i  — a/'— 6/— c;i2  =  0,  (1) 

or  .  £S=  a,    h,    g   =0,  (2) 


a, 

h, 

9 

h, 

b, 

f 

9, 

f, 

c 

is  the  required  condition. 

This  determinant  is  called  the  Discriminant  of  the  General 
Equation.  The  general  equation  of  the  second  degree  therefore 
represents  two  straight  lines  if  its  discriminant  vanishes. 

Def.  If  the  sum  of  the  exponents  of  x  and  y  is  the  same  in 
each  term  of  an  algebraic  function  (§28),  it  is  called  a  homoge- 
neous function  of  x  and  y. 

E.g.  2a?  —  4x^2/  "h  ^^2/^  —  2/^  is  a  homogeneous  function  of  x  and  y  of  the 
third  degree. 

55.*  A  homogeneous  function  of  the  nth  degree  can  be  separated  into 
n  homogeneous  linear  factors,  and  therefore,  when  equated  to  zero,  repre- 
sents n  straight  lines,  real  or  imagiriary,  through  the  origin. 

Let  the  function  be 

r+  K,y--'x  +  K,y--'x'  +  K,y"-'x'  +  •  •  •    +  K.x\     (1) 

This  is  identically  equal  to 

-■[(i)"+^'(r"+<^)"'"+^9""+-  ■  ■  +^-]-  <" 


56.]  -     THE   STRAIGHT   LINE.  •  '  76 

The  polynomial  factor  in  (2)  is  a  function  of  the  nth  degree  in 

(-),  and  therefore  has  n  roots,  real  or  imaginary.     (§  90.) 

Letm,,m2,  wig  .  .  .  »n„  be  these  robts.  Then  (2)  is  identically 
equal  to  (§  89) 

-[(i--)(l-1(i--)  •  •  •  C— )]'    ''' 

or  (^  — wii^)(2/  —  m^)(iy  —  msx)  .   .  .   (y—m^x).         (4) 

When  (4),  which  is  identically  equal  to  (1),  is  equated  to  zero, 
it  represents  the  n  straight  lines  (§63), 

y  —  m^x  =  0,    y  —  m^x  =  0,     y  —  m     =  Oj  .   .  .  y  —  m^x  =  0, 

all  of  which  go  through  the  origin. 

56.*  To  find  the  equation  of  the  lines  joining  the  origin  to  the  com- 
mon points  of 

ax'  +  2hxy  +  by'  +  2gx  ^2fy-{-c  =  0,  (1) 

and  Ix  -\-  my  =  n.  (2) 

Equation  (2)  may  be  written 

Ix  +  my  ^  ^ 
n 

Making  equation  (1)  homogeneous  and  of  the  second  degree 
by  means  of  (3),  we  get 

ax'  +  2hxy  +  bf  +  2i9x+fy)(^^)+e(^^^))=0,     (4) 

which  is  the  required  equation. 

Equation  (4),  being  homogeneous  and  of  the  second  degree,  rep- 
resents two  straight  lines  through  the  origin  (§55).  Moreover, 
the  coordinates  of  the  common  points  of  the  two  given  loci  sat- 
isfy both  equation  (1)  and  equation  (3),  and  therefore  satisfy  (4). 

For  values  of  x  and  y  which  satisfy  (3)  make  —^ — ^  equal  to 

unity,  and  therefore  give  the  same  result  when  substituted  in  (4) 
as  when  substituted  in  (1);  i.  e.,  if  they  satisfy  (1)  they  will 
also  satisfy  (4). 

Therefore  the  two  lines  (4)  pass  through  the  common  points  of 
(1)  *nd  (2). 


76  *  THE   STRAIGHT   LINE.  [57. 

In  the  same  manner  we  may  make  the  equation  of  any  curve 
homogeneous  by  means  of  (3)  and  obtain  the  equation  of  the 
straight  lines  joining  the  origin  to  the  points  common  to  the  line 
(2)  and  the  given  curve. 

Ex.  1.    Find  the  equation  of  the  lines  through  the  origin  and  the  points 

common  to 

x^  +  xy—Qx  —  Sy-]-9  =  0,    and    y-j-Sx  =  l. 

The  equation  required  is 

which  on  reduction  gives 

2x2  —  xy  —  Qy^  =  {x  —  2y){2x  +  Sy)  =  0. 
Ex.  2.    It  S  and  S^  be  two  homogeneous  functions  of  the  same  degree, 
and  K,  K^  be  two  constants,  show  that 

[S-\-K)  +  l{S'-\-K^)  =  0 
will  be  the  equation  of  the  straight  lines  through  the  origin  and  the  com- 
mon points  of  /S+  ^  =  Oand  S'+  K'=  0,  if  A  be  so  chosen  that  K  +  ?K'=  0. 

57.*  To  find  the  angle  between  the  two  straight  lines  represented  by 
the  homogeneous  equation 

aoc'^2hxy^bi/  =  0.  (1) 

Let  the  separate  equations  of  the  two  lines  be 

y  —  m^x  =  0,     and     y  —  mg^  =  0.  (2) 

Then     y' -i-2^xy -i-'^x'^  {y—m,x)(y  —  m,x)       [§55]      (3) 

=  7/2 —  (mj  -\-  m2)xy  +  m^m^x^.  (4) 

Equating  the  coefficients  of  xy  and  x^  in  (4)  gives 

mi+m2  =  — 2^,     and     m{m.2=j-.  (5) 


Whence     m^  —  m.2  =  F  (mj  -|-  m^Y  —  4m]m2  ==  h^^^  ~~  "^-       (^) 
If  (p  be  the  angle  between  the  lines  (2), 

Therefore  from  (5)  and  (6)  we  get 

.  21//1'  — a6 

tan  ^  = — — .  (7) 


58.]  THE   STRAIGHT   LINE.  77 

If  h}  —  ah  >  0,  the  lines  (1)  are  real. 

If  ^^  —  ah  =  0j  the  lines  (1)  are  coincident. 

If  A^  —  ah  <  0,  the  lines  (1)  are  imaginary. 

If  a  -|-  ^  =  0,  i.  e. ,  if  the  sum  of  the  coefficients  of  x^  and  y^  is  zero, 
isan  9?  =  00  ,  and  the  two  lines  given  by  (1)  are  at  right  angles  to 
one  another. 

Ex.  Show  that  equation  (7 )  also  gives  the  angle  between  the  two  lines 
represented  by  the  equation 

when  the  discriminant  is  zero. 

58.*  To  find  the  equation  of  the  straight  lines  bisecting  the  angles 
between  the  two  lines  given  by 

ax'-^2hxy-\-by'=0.  (1) 

Let  Yi  and  ^2  be  the  angles  which  the  lines  given  by  (1)  make 
with  the  ir-axis;  then 

y'  +  2^xy  -\-  pr'  =  {y  —  x  taji  ri)iy  ~  3C  tan  n). 

.'.     tan  ^-i-f-  tan  r2= j-,     tan  ^^  tan  ^2=  j--     [§  57,  (5).]     (2) 

Let  x  be  the  angle  that  one  of  the  bisectors  makes  with  the  ^r-axis; 
then  will 

r^Kri-^n),    or    r  =  Kri+ r2-h  ^). 
Hence  for  either  value  of  y, 

tan  2r  =  tan  (ri+Xi); 
2  tan  y     _    tan  yi-^  tan  y^  ^q\ 

1  —  tan^  r      1  —  t^ii  Ti  tan  ^'2' 
If  (xy  y)  be  any  point  on  either  bisector,  then 

taa  r  =  ^.  (4) 

X 


Substituting  (2)  and  (4)  in  (3)  gives 
2xy     _  —2h 


(5) 


2/^ —  x^      a  —  b' 
.-.     h(x'—f)  =  (a  —  b)xy  (6) 

is  the  required  equation,  since  it  is  the  relation  between  the  co- 
ordinates of  any  point  on  either  of  the  bisectors. 


78  THE   STRAIGHT   LIKE.  [58. 

EXAMPLES. 

Find  the  separate  equations  of  the  lines  represented  by  the  following 
equations,  and  determine  the  angle  between  each  pair : 

1.  a;2  — 9  =  0.  2,  xy-{-Sx~2y  =  6. 

3.  Sx''  +  2ixy-]-iOy^  =  0,  4.  a^  -  6x' -{- llx  —  6  =  0. 

5.  4a;2  — 24c2/  +  lly2  =  0.  6.  ix''-[-20xy  +  9y^  =  0. 

7.  2x''-\-Sxy  —  2y^  =  0.  8.  y^-{-xy^  —  lix''y  —  2i3c^  =  0. 

9.  x''  +  2xysece-\-y^  =  0.  10.  x"" -\-2xy  cot2d —  y^  =  0. 

11.  x''-\-2xycsc2d-}.y'  =  0.  12.  2/^  cot^  0  +  2x2/ +  jc^  sin^  ^  =  0. 
What  loci  are  represented  by 

13.  y*=16aV.  14.  (x2+ 2/2)2— 4rV=0. 

15.  ^2  _j_  (a.3_  2^)2/ —  x^  =  C.  16.  2/*  +  (x  — a^)2/2  — x*  =  0. 

17.  Find  the  equations  of  the  bisectors  of  the  angles  between  the  pairs 
of  lines  given  in  examples  3,  5,  6,  7,  9,  and  10. 

18.  Show  that  the  two  straight  lines 

(a:^ —  y^)  sin  2<j)  +  2{x  sin  ^  —  y  cos  (py  cot  6  =  2xy  cos  2<j> 
include  an  angle  6. 

Show  that  the  following  equations  represent  straight  lines;  find  their 
point  of  intersection  and  the  angle  between  them :     [Solve  for  x  ory.'] 

,19)  x2  +  3x2/  +  22/2  — 3x  — 32/  =  0. 

20.  lOx'  — 13x2/  — 32/''  +  16x  +  102/  — 8  =  0. 

21.  x2  — X2/  — 22/^  — X  — 42/  — 2  =  0. 

22.  2x2  +  5x2/  — 32/'  +  6x  — 102/  — 8  =  0. 
■''23)  2x2  — 3x2/  — 22/'  — lOx  — 102/— 12  =  0. 

24.  4x2  +  12x2/  +  92/^  —  18x  — 272/ +18  =  0. 

Find  the  respective  values  of  A  for  which  the  following  equations  repre- 
sent line  pairs : 

25.  x2  — 41/2  — 2x  +  82/  +  ^=0. 

26.  12x2  — X2/+A2/2  +  2X  +  72/  — 2  =  0. 

27.  Ax2  — 7x2/  — 52/2- 14x +  322/  — 12=0. 

28.  x2  — 4x2/  +  42/2  +  3x  — 62/  +  ;i  =  0. 

29.  Ax2/  +  32/2  —  2x  — 122/ +  12  =  0. 


58.]  THE  STRAIGHT  LLNE.        '  79 

30.  a;'  +  5a^  +  6y'  +  ^a:  — 122/  — 48  =  0. 

31.  i2x^-{-2?^xy—3y^-^iOx-\-25y  —  28  =  0, 

32.  6x'-\-xy—15y^-{-Xx  +  50y  —  ^0  =  0. 

33.  x»  — 6x2/  +  ^3/'  — 8x  +  8X2/+16  =  0. 

34.  x" -{- hey -\'dy'  —  Sx  —  i^y  + 16  =  0. 

35.  "What  are  the  conditions  that  the  equations 

ax''-}-by^-{-gx-{-gy  =  0    and    ay' -\- hxy -\- gx -^fy  =  0, 
may  each  represent  a  pair  of  straight  lines  ? 

36.  Find  the  equations  of  the  straight  lines  passing  through  the  origin 
and  the  points  of  intersection  of 

(1)  2/2  =  4a;    and    2x-\-y  =  12. 

(2)  (x  — 6)' +(2/  — 2)2  =  20    and    y  +  Sx  =  iO. 
What  is  the  angle  between  the  last  pair  of  lines? 

37.  Find  the  angle  between  the  lines  which  join  the  origin  to  the  com- 
mon points  of 

3a;2  — a;2/  — 22/2  +  10x  +  8  =  0    and    42/— 3x  =  10. 

38.  Show  that  the  lines  through  the  origin  and  the  points  of  intersec- 
tion of 

x^  -|-  2/^^  =  2    and    y  =  mx  +  2 

are  at  right  angles  if  m  =  =b  v/3. 

39.  Show  that  the  straight  lines  joining  the  origin  to  the  points  of  in- 
tersection of  the  straight  line  x  —  y  =  2  and  the  curve 

22/2  —  2x2/ —  3x2  — 4x  + 42/ +  4  =  0 

make  equal  angles  with  the  axes. 

40.  Prove  that  the  angle  between  the  lines  joining  the  origin  to  the 
points  common  to  the  straight  line  x-\-2y  =  1  and  the  curve 

102/2  —7x2/  +  4x2 _2x  —  42/  +  l  =  Ois  tan-i ^. 

41.  Find  the  equation  of  the  straight  lines  passing  through  the  origin 
and  the  common  points  of 

1  +  1  =  1    and    x2+2/2  =  16. 

42.  Show  that  the  lines  passing  through  the  origin  and  the  points  com- 
mon to 

4a2 


^  +  ^,  =  1    and    5(x2  +  2/2)  =  8a2 


are  perpendicular  to  each  other. 


80  '  THE   STRAIGHT   LINE.  [59. 


Oblique  Axes. 

59.     To  find  the  equation  of  a  straight  line  referred  to  axes  inclined 
at  an  angle  lo. 


Let  ABPhQ  any  line  meeting  the  ?/-axis  at  a  distance  h  from 
the  origin,  and  making  an  angle  y  with  the  a^'-axis. 

Draw  PQ  parallel  to  the  ^/-axis  and  OR  parallel  to  the  given 
line  ABP. 

Let  P(^x,  y)  be  any  point  on  the  line  ABP-,  then 

OQ  =  X,     and     QR  =  QP—  RP=y  —  b. 

Since  Z  ORQ  =  Z,ROY  =  co  — y,  we  also  have 

y  —  b  _  QR  _  sin  Q  OR  _        sin  ^  " 

X     ~  0Q~  sin  ORQ~~  sin  (oj — y)' 

which  is  the  required  equation. 

T   X*  sinr  tan  y 

Let  m  =  - — -. — - — r  =  ^ 7 •  (2) 

sm  («>  —  y)      sm  w  —  cos  (o  tan  y 

r^-,  .  ^  sin  o) 

Then  t&n  y  = -—, ,  (3) 

1  -f  m  cos  w 

and  equation  (1)  becomes 

y  —  mx  H-  6,  (4) 

which  in  oblique  coordinates  represents  a  straight  line  inclined 

to  the  a?-axis  at  an  angle  tan~M :; — ■ 1. 

°  \1  -\-  m  cos  lof 


61.]  THE   STRAIGHT   LINE.  81 

60.  Some  of  the  investigations  in  the  preceding  sections  of 
this  chapter  apply  to  oblique  as  well  as  to  rectangular  axes.  Let 
the  student  show  that  this  is  true  of  the  following  equations: 

f+f  =  l,  [(!),§  41] 

y_y^=,n{x  —  x,),  [(3),  §46] 


y  —  yi  ^  x  —  oc, 
1/2— y  I    0-2— jci 


[(3):  §  47] 


61.*     To  find  the  angle  between  two  straight  lines  ivhose  equations^ 
referred  to  axes  inclined  at  an  angle  w,  are 

y  =  mx  -f-  h     and     y  =  m'x  -\-  h'. 

If  Y  and  y'  are  the  angles  which  these  lines  make  respectively 
with  the  a:;-axis,  then  [§  59,  (3)] 

m  sin  o)  ^         ,  m'  sin  (o 

tan  ;- =  — ,     tan /-'=-— .  (1) 

1  -f  m  cos  lo'  1  -\-  m'  cos  (o  ^   ^ 

m  sin  (o  m'  sin  w 


„.,  ,       .  ,.  1  4-  m  cos  w       1  +  m'  cos  «>  .^. 

Whence    tan  {y  —  r^  — : — r— : •      (2  ) 

m  sm  o)  m  sm  </> 


1  +  m  cos  oi  '  1  -\-m'  cos 


iU 


(m  —  m')  sin  cu 

,'.    tan  s^  =  ^    ,    .      , — y^ . „  (S) 

1  -(-  (m  -f-  ??i')cos  oj  -\-  mm'  ^   ^ 

where  <p  =  r  —  r\  ^^^  angle  between  the  lines. 
The  two  given  lines  are  parallel  if  m  =  m'. 
They  are  perpendicular  to  one  another  if 

1 +(m +  m')  cos  <« -j- mm' =  0.  (4) 

If  the  equations  of  the  given  Unes  are 

Ax-\-Bi/-^C  =  0    and     A'x -^  B'y -^t  C'=  0, 

A  A' 

then  m  =  — „     and     m'= — ^^. 

Substituting  these  values  in  (3)  we  have 

-  (^'^--^^Qsin  io 

*^''^~  AA'-]-BB'—(AB'+A'B)co&<o'  ^^^ 


82  THE   STRAIGHT   LINE.  [62. 

The  lines  will  be  parallel  if  A'B  —  AB'=0. 
They  will  be  perpendicular  to  one  another  if 

AA'^BB'— (AB'-^  A' B)  cos  io  =  0.  (6) 

62.  To  find  the  equation  of  a  straight  line  in  terms  of  p,  the  per- 
pendicular upon  it  from  the  origin,  and  the  angles  a,  /S  which  p  makes 
with  the  axes. 


Let  AB  be  the  given  line  and  OR  the  perpendicular  on  it  from 
the  origin. 

Let  P(a;,  y)  be  any  point  on  AB.     Dr^w  QP  parallel  to  the 
t/-axis,  and  NP  parallel  to  OR. 

Then      lY0R  =  lNPQ  =  >3,     OQ  ^  sr,     QP=y. 

Projecting  OQ  and  QP  on  OR  gives 

OQ  cos  a  -\-  QP  cos  /5  =  OR. 

.'.      X  cos  a  -\-  y  cos  i3  =  p,  (1) 

is  the  required  equation. 

Let  r  =  Z  ^BA  =  Z  XQM=  a  -f  90°. 

Then  /  PQM  =  90°—  ,3  =  r  —  <^- 

. • .     cos  a  =  sin  ^     cos  ;?  =  sin  (^  —  (o)—  —  sin  (w  —  ^), 

and  the  equation  (1)  of  the  line  may  be  written 

X  sin  y  —  y  sin  (o)  —  ^)  r=  p.  (2) 

Equation  (1),  or  (2),  is  called  the  normal^  or  distance  form  of  the 
equation  of  the  straight  line  when  the  axes  are  oblique. 


63.]  THE   STRAIGHT   LINE.  83 

The  conventions  as  to  the  direction  of  p  and  AB,  and  the  posi- 
tive and  negative  sides  of  AB,  are  the  same  as  in  §  42. 

63.      To  change  the  general  equation 

Ax^Bu+C  =  0,  (1) 

referred  to  oblique  axes,  to  the  distance  fann 

X  cos  a  -\-  y  cos  /5  — p  —  0.  (2) 

Since  (1)  and  (2)  represent  the  same  line  their  first  members 
are  identically  equal,  and  therefore  their  coefficients  are  propor- 
tional, i.  e. 

cos  a  _  cos  /?    _  — p 

'~^~~~B~~~C~'  ^^^ 

p^    _  COS^a  _  COSV  _  2  COS  a  COS  /5  COS  (a  -|-  i?) 

•*•     'C^~~¥~~~W~  ~        2AB  COS  (a  +  /5) 

_  COS^  a  -\-  COS^  /?  —  2  COS  a  COS  /5  COS  (a  -|-  /?) 

~~^  ^-'4- jB^— 2^^  COS  («  +  /?)  •   ^^^ 

But 

COS^  a  +  cos^3  —  2  COS  a  COS  /3  COS  (a  -f  /5)  =  sin^(a  -f  /9)  =  sin^w, 
since  a  -f  /5  =  a>,  the  angle  between  the  axes, 
cos  a  _  COS  /5  _  — ^  _  sin  w 


'  '        A  B 

Whence  cos  a 


C         \/A'^B'—2AB 

cos  io 

A  sin  lo 

VA'-{-B'—2ABqoqu,' 

B  sin  (t) 

VA^-\-B^—2ABqo^<o' 

—  0  sin  w 

(5) 
(6) 


fiOH  /9  =  (-7'. 

(8) 


and  „  , 

VA^-^B^—2ABco^io 


Substituting  (6),  (7),  and  (8)  in  (2)  gives 

{Ax  -\-  By  -\-  C)  sin  (o 


VA'-^-B'— 2 AB  cos  i 
which  is  the  distance  form  of  the  general  equation  (1) 


0,  (9) 


84  THE   STRAIGHT   LINE.  [64. 

64.  To  find  the  perpendicular  distance  of  a  given  point  (Xi,  y^)  from 
the  line  whose  equation  is 

X  COB  a  -\-  y  cos  /5  — p  =  0,  (1) 

ir  sin  ^  —  ysm((o — y) — P  =  0,  (2) 

or  Ax-^By-^C  =  0,  (3) 

where  (o  z=  a  -\-  ^  is  the  angle  between  the  axes. 

The  demonstration  given  in  §  50  applies  also  when  the  axes  are 
oblique.     The  required  results  are,  respectively, 

Xi  cos  «  +  2/i  COS  /5  — p,  (4) 

Xi8inr  —  y^8in{io  —  y)—p,  (5) 

,  (Axi  +  ^Vi  +  C)  sin  o) 

and  y  (6) 

\/A'-{-B'— 2 AB  COS  CO  ^  ^ 

Observe  that  formulse  previously  found  independently  for  rect- 
angular axes  can  now  be  derived  from  those  here  obtained  for 
oblique  axes  by  simply  putting  a>  =  90°.     (  Cf.  §§  7  and  10.) 

EXAMPLES. 

1.  The  axes  being  inclined  at  an  angle  of  60°,  find  the  inclination  to  the 
X-axis  of  the  straight  lines 

2/-X-3,    (l/3-l)2/  =  2x+(v^3  +  l),    2y  +  x  =  L 

2.  If  w  =  120°,  find  the  angle  between  the  two  lines 

(1)  y-\-3x  =  S    and    %  =  x  +  8, 

(2)  y=3x  —  2    and    2y-\-x  =  i. 

3.  Show  that  when  the  angle  between  the  axes  is  w,  the  angle  between 
the  two  lines 

y  —  mx  =  0  and  my-\-x  =  0  is  tan~M     ,^  *"    tanojj. 

4.  Prove  that  the  straight  lines 

y  =  x-\-c    and    y-\-x  =  b 
are  at  right  angles,  whatever  be  the  angle  between  the  axes. 

5.  If  the  lines 

Sy^2x  =  S    and    ly-{-Sx^U  =  0 

are  at  right  angles,  what  is  the  value  of  w  ? 

6.  Show  that  the  two  points  (xi,  yi)  and  (X2, 1/2)  are  on  the  same  side  or 
on  opposite  sides  of  the  line  Ax  -\-  By  -f-  C  =  0  according  as  Axi  +  Byi  -f  C 
and  ^X2  -f  By2  +  C  have  the  same  sign  or  opposite  signs,  the  axes  being 
oblique. 


64.]  THE  STRAIGHT  LINE.  85 

7.  Find  the  equations  of  the  bisectors  of  the  angles  between  the  follow- 
ing lines  when  w  =  60° : 

(1)  y  =  x-{-2    and    y -\-2x-\-5  =  0. 

(2)  x  +  2/  +  3  =  0    and    22/-|-(l -f  v/6)x  +  4  =  0. 

(3)  3x-\-Qy-\-S^0    and    6x-|- 3^  =  10. 

8.  If  w  =  30°,  find  the  equation  of  the  line  through  the  point  (2,  3),  and 
(1)  parallel  to,  (2)  perpendicular  to  the  line  3y  —  2x. 

9.  Find  the  length  of  the  perpendicular  drawn  from  the  point  (2,  —  4) 
upon  the  line  3x -\- 6y -\- ii  =  0,  when  w  =  60°. 

10.  Find  the  equation  to,  and  the  length  of  the  perpendicular  drawn 
from  the  point  (—  3,  —  2)  to  the  line  ix-\-3y  =  Q,  when  w  =  60°. 

11.  Prove  that  the  equation  of  the  line  which  passes  through  the  point 
(Xi,  2/1 )  and  is  perpendicular  to 

(1)  2/-0, 

(2)  x  =  0, 

(3)  a;sin7  +  2/sin(>'  — w)=p 
is,  respectively, 

(1)  (X  — x,)  +  (2/  — 2/i)coso>  =  0, 

(2)  (X— x,)cosw  +  2/— 2/1  =0, 

(3)  (x  — xi)  cos7H-(2/  — 2/i)cos(7  — 6;)  =  0. 

12.  Show  that  the  equation  of  the  line  through  the  point  (Xi,  2/1)  per- 
pendicular to  the  line 

2/  =  mx  -\-  b 

,         .^^  l  +  mcosw 

may  be  written  y-.y,=-^^^-^--—.(x-x,). 

13.  Show  that  the  lines 

X  -f  2/  cos  0)  =  b  cos  w    and    x  cos  u-{-y  =  b 
are  perpendicular  to  the  axes  of  x  and  2/ respectively. 

14.  If  2/  =  wx  -f-  b  and  y  =  m^x  -\-  b^  make  equal  angles  with  the  x-aiis 
and  are  not  parallel,  prove  that 

m  -|-  m^-j-  2mm'  cos  w  =  0. 

15.  Find  the  equations  of  the  sides  of  a  regular  hexagon  when  two  of 
the  sides  which  meet  in  a  vertex  are  the  axes  of  coordinates. 

16.  PA  and  PB  are  the  perpendiculars  upon  the  axes  from  the  point 
P^ttf  b);  it  0)  be  the  angle  between  the  axes,  prove  that 

AB  =  sin  to  \a^  +  6*  +  2a6  cos  w. 


86  THE   STRAIGHT   LINE.  [64. 

17.    Show  also  that  the  length  of  the  perpendicular  from  P  on  AB  in 
Ex.  16  is 

ab  sin^  o 


Va^-^b''-\-2abcosu 
and  that  its  equation  is  ax  —  by  =  a'^  —  b"^. 

18.  From  each  vertex  of  a  parallelogram  a  perpendicular  is  drawn  upon 
the  diagonal  which  does  not  pass  through  that  vertex,  and  these  are  pro- 
duced to  form  another  parallelogram ;  show  that  its  diagonals  are  perpen- 
dicular to  the  sides  of  the  first  parallelogram  and  that  they  both  have  the 
same  centre. 

19.  If  the  axes  be  inclined  at  an  angle  w,  show  that  the  equation  of  a 
straight  line  through  (xj,  2/j)  making  a  given  angle  7  with  the  x-axis  may- 
be written 

[x  —  xi)  sin  7  —  {y  —  2/1)  sin  (w  —  y)  =  0, 

and  also  that  this  equation  is  in  the  distance  form. 

20.  Find  the  angle  between  the  lines 

ax'-\-2hxy~\-by'^=Q, 

when  the  ax^s  are  inclined  at  an  angle  w.    Show  also  that  these  lines  are 
at  right  angles  to  one  another  if 

a-\^b —  2h  cos  w  =  0. 


Examples  on  Chapter  III. 

1.  What  are  the  loci  of  the  following  equations? 

(1)  x'-^axy  =  0.  (2)  x^  —  xy''  =  ^. 

(3)  x^  +  2/'=0.  (4)  x"*  — 2/3=0. 

(5)  aV  — 5V=0.  (6)  aV  +  by=:0. 

(7)  (x^  — 1)(2/'-^  — 4)=0.  (8)  iax  +  byy^c\ 

(9)  2/2  — (X  — a)'^=0.  (10)  (X  — a)2  +  (2/  — b)'-^  =  0. 

(11)  (x  —  ay—{y  —  by=Q.  (12)  x''  —  x'y^xy''  —  y'=Q, 

(13)  ^  =  asec(^  — a). 

2.  Find  the  angle  between  the  two  lines  3x  =  42/  +  7  and  hy  =  12x  +  6; 
also  the  equations  of  the  two  lines  which  pass  through  the  point  (4,  5)  and 
make  equal  angles  with  the  two  given  lines. 

3.  Find  the  length  of  the  perpendicular  from  the  origin  upon  the  line 
passing  through  the  points 

(a  cos  o,  a  sin  «)    and    (a  cos  /5,  a  sin  ft). 


64.]  THE   STRAIGHT   LINE.  87 

4.  What  is  the  equation  of  the  line  through  the  intersection  of  the  two 
straight  lines 

bx-]-ay  =  ab    and    y  =  ma;, 

and  perpendicular  to  the  former  ? 

5.  Prove  that  the  equation  of  the  two  straight  lines  which  pass  through 
the  origin  and  make  an  angle  a  with  the  line  y-\-x  =  0i8 

x^-\-2xysec2a-\-y^=0. 

.    6.    Find  the  perimeter,  altitudes,  and  area  of  the  triangle  whose  vertices 
are  at  the  points  (3,  5),  (7,  9),  (9,  11). 

7.  If  p  and  p^  be  the  perpendiculars  from  the  points  {±:Va^  —  b'S  0) 
upon  the  line  —  cos  ^  +  ^  sin  ^  =  1,    , 

prove  pp^  =  b^. 

8.  Find  the  equations  of  the  sides  of  the  square  of  which  the  points 
(2,  -^  3)  and  (6,  5)  are  two  opposite  vertices. 

9.  Show  that  the  equation 

3/3  —  x^  +  3x1/(2/  —  x)  =  0 
represents  three  straight  lines  equally  inclined  to  one  another. 

10.  Prove  that  the  equation 

2/2 (cos  a-\-  T''  3  sin  «)  cos  a  —  x2/(sin  2«  —  V  3  cos  2a) 

-\-  x2(sin  a  —  i '3  cos  o)  sin  a  =  0 
represents  two  lines  inclined  at  an  angle  of  60°  to  each  other. 

11.  For  what  value  of  m  vnll  the  lines 

bx-{-ay  =  abf    ax-\-by  =  abj    y  —  mx 
meet  in  a  point  ? 

12.  Show  that  the  lines 

y  =  rriix  +  5,,    y=  niox  +  60,    y  =  mgx  -f-  63 

will  meet  in  a  point  if 

ma  —  nil  _  &3  —  61 
rrio  —  mi      &2  —  &i* 

13.  From  a  point  P  perpendiculars  PM  and  PN  are  drawn  upon  two 
fixed  lines  which  are  inclined  at  an  angle  o  and  meet  in  O.  Take  the  two 
fixed  lines  as  axes  of  coordinates  and  find  the  locus  of  P  if 

(1)    OM  +  ON  =  2c.  (2)     0M—0N=2c. 

(3)    PM-\-PN=2c,  (4)    PM—PN  =  2c. 

(5)    MN  =  2c.  Ans.  to  (5) .    x^  +  2xy  cos  u-\-y^  =  4c^  csc^  w. 


88  THE   STRAIGHT   LINE.  ^  [64. 

14.  Find  the  points  of  intersection  of  the  loci 

pcos(6  —  ^-)=a    and    pcos(d — i7r)=a> 

15.  also  of         p  cos  (0  —  ^rr)  =  fa    and    p  =  a  sin  6, 

16.  OA  and  OB  are  two  fixed  straight  lines,  A  and  B  being  fixed  points; 
P  and  Q  are  any  two  points  on  these  lines  such  that  the  ratio  AP :  BQ  is 
constant.    Show  that  the  locus  of  the  middle  point  of  PQ  is  a  straight  line. 

17.  PM  and  PN  are  perpendiculars  from  a  point  P  on  two  fixed  straight 
lines  which  meet  in  O;  MQ  and  NQ  are  drawn  parallel  to  the  fixed  lines  to 
meet  inQ;  prove  that  if  the  locus  of  Pis  a  straight  line,  the  locus  of  Q 
will  also  be  a  straight  line. 

18.  ABCD  is  a  parallelogram.  Taking  A  as  pole  and  AB  as  initial  line, 
find  the  polar  equations  of  the  four  sides  and  two  diagonals. 

19.  A  straight  line  moves  so  that  the  sum  of  the  reciprocals  of  its  in- 
tercepts on  two  fixed  intersecting  lines  is  constant;  show  that  it  passes 
through  a  fixed  point. 

20.  The  distance  of  a  point  (xi,  2/1)  from  each  of  two  straight  lines, 
which  pass  through  the  origin,  is  d;  prove  that  the  two  lines  are  given  by 

21.  Show  that  the  six  bisectors  of  the  angles  formed  by  the  lines 

X  cos  ci -|- 3/ sin  oj  =pi, 

X  cos  «2  +  2/  sin  a.,  =  po,    x  cos  «3  -|-  2/  sin  a^  =  ps, 

meet  in  sets  of  three  in  four  different  points.    What  are  these  four  points  ? 

22.  Prove  that  the  three  altitudes  of  a  triangle  meet  in  a  point. 

23.  Prove  that  the  three  perpendicular  bisectors  of  the  sides  of  a  tri- 
angle meet  in  a  point. 

24.  Find  the  area  of  the  triangle  formed  by  the  lines 

2/  +  3x  =  6,    2/=2x-4,    y  =  ix-^S. 

25.  Show  that  the  area  of  the  triangle  formed  by  the  lines 

y  =  rriix  -\-bi,    y  =  rajx  -\-  ho,    and    x  =  0, 

"  wii  —  m2 

26.  Show  that  the  area  of  the  triangle  formed  by  the  lines 

y  —  TtiiX  -f-  61,    2/  =  w^a;  +  62,    and    y  =  m^x  +  63 
i,  [-(b.-b.)'      (6.-6ay      (j,:^)'-|     (Use  Ex.  25.) 


64.]  THE   STRAIGHT   LINE.  89 

27.  What  is  the  area  of  the  triangle  whose  sides  are  the  lines 

3a;  +  %  +  12  =  0,    2x  +  y  =  4,    5x—3y  =  i5? 

28.  Find  the  equation  of  the  pair  of  lines  joining  the  origin  to  the  in- 
tersections of  the  straight  line  y  =  mx  +  h  and  the  circle  ic^  +  2/^  =  r^. 

Show  that  these  lines  will  be  at  right  angles  if 

262  =  r2(l-f  m^); 
and  coincident  if  6^  =  r\\-\-  m^). 

29.  Prove  that  the  straight  lines  joining  the  origin  to  the  points  of  in- 
tersection of  the  line  hx-\-ay  —  2db  with  the  circle' 

(x  — a)2+(2/  — 5)2  =  r2 

will  be  at  right  angles  if  a^-\-h'^  =  r'^, 

30.  Show  that  the  two  straight  lines  joining  the  origin  to  the  other 
points  of  intersection  of  the  two  curves 

ax"  +  2hxy  +  hy""  -f  2srx  =  0 

and  a'x^  +  2Wxy  +  h'y''  +  2g'x  =  0 

will  be  perpendicular  to  one  another  if 

g^(a  +  b)  =  gia^-^b'). 

31.  Prove  that  the  angle  between  the  two  lines  joining  the  origin  to  the 
intersections  of  the  line  2y  =  Sx-\-2  with  the  curve 

10x2  —  14x2/ +  32/2  —  5a; -I- 22/ —  2  =  0  is  tan-i  I . 

32.  Show  that  bx^  —  2hxy  +  ciy^  =  0  represents  two  straight  lines  at 
right  angles  respectively  to  the  two  straight  lines 

ax2 +  2/1X2/+ 62/' =0. 

33. .  If  the  pairs  of  straight  lines 

x^  —  2pxy  —  y^  =0    and    x^  —  2qxy  —  y"^  =  0 

be  such  that  each  pair  bisects  the  angles  between  the  other  pair,  prove 
that  pq=  —  1- 

34.  Find  the  locus  of  the  vertex  of  a  triangle  which  has  a  given  base 
and  a  given  difference  of  base  angles. 

35.  The  product  of  the  perpendiculars  drawn  from  a  point  P(x%  y')  on 
the  lines 

X  cos  0  -\-y  %mO  =  a    and    x  cos  ^  +  2/  sin  <»  =  a 

is  equal  to  the  square  of  the  perpendicular  drawn  from  P  on  the  line 

X  cos  ^(^  +  0)  +  2/ sin  =]  (6/ +  ^)  =  a  cos  J(^  —  0), 
Show  that  the  equation  of  the  locus  of  P  is 


90  THE   STRAIGHT   LINE.  [64. 

36.  Prove  that  the  straight  lines 

ax^H- 2/1X2/ +  62/2=0, 

make  equal  angles  with  the  x-axis  it  h  =  a  cos  w,  the  axes  being  inclined 
at  an  angle  w. 

37.  If  w  be  the  angle  between  the  axes,  show  that  the  lines  given  by  the 
equation 

x^  -f-  2x2/  cos  w  -|-  2/^  cos  2w  =  0 

are  at  right  angles  to  one  another. 

38.  Prove  that  the  general  equation 

ax'  +  2/1X2/  +  by'  -\- 2gx -^  2fy  +  c  =  0 
represents  two  parallel  straight  lines  if 

h^=ah    and    bg'=ap. 
Prove  also  that  the  distance  between  them  is 


\a[a  +  h) 


39.  Show  that  the  product  of  the  perpendiculars  from  the  point  (x^,  y^) 
upon  the  two  lines 

ax2-f  2/1X2/ +  62/2=0 

ax^2_j_  2hx'y^-\-  by^^ 

40.  Show  that  the  pair  of  lines  given  by 

ax'  +  2/1X2/  +  62/2  +  Hx'  -\-y^)=0 
is  equally  inclined  to  the  pair  given  by 

ax2  +  2/1X2/  +  62/2  =  0.  (Use  §  58.) 

41.  Show  also  that  the  pair 

a2x2  +  2/i(a  +  b)xy  +  bY  =  0 
is  equally  inclined  to  the  same  pair. 

42.  If  the  general  equation 

ax'  +  2/1X2/  +  62/2  +  2gx  +  2/2/  +  c  =  0 

represents  a  pair  of  straight  lines,  prove  that  the  equation  of  the  other  pair 
of  lines  meeting  the  axes  in  the  same  points  is 

ax2  +  2(^  - /i)x2/ +  62/2  +  2srx  +  2/2/ +  c  =  0. 

43.  Prove  that  the  three  lines  represented  by  the  equation 

x(x2  —  Sy'')=  myiy^  —  3x2) 
make  equal  angles  with  one  another. 
(Hint.    Show  that  the  polar  equation  is  m  tan  3^  +  1=0.) 


64.]  THE  STRAIGHT   LINE.  91 

44.  Show  that  the  condition  that  two  of  the  lines  represented  by  the 
equation 

Ax^  -f  SBx'y  +  SCxy^  -\-Dy^=0 

may  be  at  right  angles  is 

A'  4-  SAC  +  3BD  -\-D^=0. 
SUG.    The  given  equation  must  be  equivalent  to 

{Ix  +  my){x'  +  ?ixy  -  y')  =  0.) 

45.  Show  that  ( — j  +  ( ^\  —  Q  =  0  represents   two    pairs  of 

perpendicular  lines  through  the  origin. 

46.  If  2  =  -  —  ^,  show  that 

y      X 

z" -\- az"-'' -\- bz»-^ -\-  .  .  .  kz-\-l=0 

represents  n  pairs  of  perpendicular  lines  through  the  origin. 

47.  Show  that  x* -f  4(x-  —  y^)xy  —  x^y'^  -\-y^—^  represents  two  pairs  of 
perpendicular  lines  through  the  origin. 

48.  Show  that  the  equation 

a(x*+  2/*)  —  iLbxyix"  —  y"")  -\-  ^cx^y''-  0 

represents  two  pairs  of  straight  lines  at  right  angles  to  one  another,  and 
that  the  two  pairs' will  coincide  if 

262=a2+3ac. 


CHAPTER  IV. 

TRANSFORMATION  OF  COORDINATES,  OR  CHANGE  OF 

AXES. 

65.  The  formulae  for  changing  an  equation  from  rectangular 
to  polar  coordinates  and  vice  versa  have  already  been  found  in  §  6, 
and  their  usefulness  amply  illustrated.  Moreover,  the  equation 
of  a  curve  in  any  system  of  coordinates  is  sometimes  greatly  sim- 
plified by  referring  it  to  a  new  set  of  axes  of  the  same  system. 
Hence,  it  is  also  desirable  to  be  able  to  deduce  from  the  equation 
of  a  curve  referred  to  one  set  of  axes  its  equation  referred  to  an- 
other set  of  axes  of  the  same  system.  Either  of  these  operations 
is  known  as  a  Transformation  of  Coordinates,  or  Change 
of  Axes. 

The  equations,  which  express  the  relations  between  the  two 
sets  of  coordinates  of  the  same  point,  and  by  means  of  which  these 
operations  are  performed,  are  called  Formulae  of  Transforma- 
tion. 

Change  of  Axes  in  Cartesian  Coordinates. 

SQ.  To  change  from  one  set  of  rectilinear  axes  inclined  at  an  angle 
lo  to  any  other  set  inclined  at  an  angle  to'. 

Let  OA"  and  OF  be  the  positive  directions  of  the  original  axes, 
O'X'  and  O'Y'  the  positive  directions  of  the  new  axes;  let 
XOY=  ft>,  and  X'0'Y'=  w',  be  positive  angles  less  than  tt. 

Let  O'X'  meet  OX  and  OYin  A  and  B  respectively. 

Let  Z  XAX'=  d,  then  /  X'J5F=  io—O. 

Let  h,  h  be  the  coordinates  of  the  new  origin  0'  referred  to  the 
original  axes. 

The  equation  of  any  line  referred  to  the  new  axes  may  be  writ- 
ten in  the  distance  form.  [(2),  §  62] 

x'^iny' — f/' sin  (w' — y'^^p'^  (1) 

where  y'  is  the  angle  the  line  makes  with  O'X',  p>  is  the  distance 
from  0'  to  the  line,  and  the  primes  are  used  to  denote  that  the 
equation  is  referred  to  the  new  axes. 


66.] 


CHANGE   OF   AXES. 


93 


For  the  line  OX  the  positive  direction  of  p'  is  downward  (§42), 
while  the  positive  direction  of  k  is  upward;  hence  ^'  and  k  have 
the  same  sign.  For  OF  the  positive  direction  of  p'  is  toward  the 
rights  or  the  same  as  the  positive  direction  of  h;  hence  j?'  and  h 
have  opposite  signs. 


Therefore,  for  all  relative  positions  of  the  two  pairs  of  axes  we 
have,  since  sin  w  is  positive, 

for  OX,  r'^iX'AX=  —  iXAX'=  —  d, 

p'=  Dist.  from  0'  to  OX  =  A;  sin  w  ; 
for  OY,  f=AX'BY=  ^  —  < 

p'=I)ist.  from  0'  to  0Y=  —  ^  sin  <y. 

Therefore  the  equations  of  OX  and  0  Y  referred  to  the  new 
axes  are,  respectively,  from  ( 1 ) 

x'  sin(—  0)  —  y'  sinloj'  —(—  O)]  r=  A;  sin  w,  (2) 

and      x'  sin  (w  —  0)  —  ?/'  sin [«>'  —  (w  —  ^)]  =  —  h  sin  w.  (3) 

When  (2)  and  (3)  are  written  in  the  form 

x'  sin  0  -{-  if  sin  (w'-f  0)  -\- k  eiJKo  =  0,  (4) 

and        x'  sin  (w  —  0)  —  y'  sin  (w'  —  m  -\-  0)  -^  h  sin  w  =  0,  (5) 


94  CHANGE   OF   AXES.  [66. 

the  positive  sides  of  OX  and  0  Y  with  reference  to  these  equa- 
tions (§50)  are  the  same  as  their  positive  sides  when  they  are 
considered  as  the  original  axes  of  coordinates. 

Let  P  be  any  point  whose  coordinates  are  x,  y  referred  to  OX 
and  OF,  and  x',  y'  referred  to  O'X'  and  0'  Y\ 

Draw  FM  and  PN  perpendicular  to  OX  and  0  F,  respectively. 

Then  from  (5)  and  (4)  we  get  [(5),  §  64] 

NP  —  irsin  w  =  x'  sm  (w  — 6)  —  ysin(w'  —  co  -\-  6)-\-  /isin  w,    (6) 

MP  =  y  ^in  w  —  x'  sin  0  -\-  y'  sin  («>'+  <^)  +  ^  sin  lo.  (7) 

Whence 

X  =  [x'  sin  («>  —  6)  —  y'  sin  {m'  —  «>  -f  ^)]  esc  (o  -\-h,^ 

y  —  [x'  sin  0  ^  y'  sin  («>'+  0)']  esc  w  -|-  A;.  J 

These  formulae  give  the  values  of  the  old  coordinates  of  any 
point  in  terms  of  the  new  coordinates ;  and  if  these  values  be 
substituted  in  a  given  equation,  the  result  will  be  the  equation 
of  the  same  curve  referred  to  the  new  axes. 

When  the  origin  remains  the  same,  and  only  the  direction  oj  the  axes 
is  changed,  ^  =  A;  =  0,  and  we  have 

X  =  \_x'  sin  (o)  —  0)  —  y'  sin  (w'  —  w  -{-  0)'\  esc  lo,  ^ 

y  =  [a?'  sin  0  -\-  y'  sin  («>'+  <?)]  esc  «>.  J 

These  general  formulae  (8)  and  (9)  are  rarely  used  in  the  man- 
ner suggested  above,  but  some  of  the  forms  which  they  take  in 
certain  particular  cases  are  of  great  importance. 

To  change  the  origin  to  the  point 
(h,  k)  without  changing  the  direc- 
tion of  the  axes. 

Since  in  this  case  the  new  axes 
are  parallel  respectively  to  the 
old, 

and  ^  =  0. 


o 

Substituting  these  values  in  (8)  we  get 

X  = 

y 


=  x'-{-h,  j 


(10) 


60.] 


CHANGE  OF   AXES. 


96 


As  these  equations  are  independent  of  w,  they  hold  for  both 
rectangular  and  oblique  coordinates. 

Hence  to  find  what  a  given  equation  becomes  when  the  origin 
is  moved  to  the  point  {h,  k),  the  new  axes  being  parallel  to  the 
old,  substitute  x'-\-  h  for  x  and  y'-\-  h  for  y.  After  the  substitu- 
tion is  made  we  can  write  x  and  y  instead  of  x'  and  y'\  so  that 
practically  this  transformation  is  effected  by  simply  writing  x  -\-h 
in  the  place  of  x,  and  y  -{-  k  in  the  place  of  y. 

To  turn  a  set  of  rectangular  axesiy' 
through  an  angle  0  without  chang- 
ing the  origin. 

In  this  case 

o}  =  io'=  90°, 

and  the  general  formulae  (9)  re- 
duce to 


x  =  x'  cos  0  —  y'  sin  Oj 
y  —  x'  sin  0  -{-  y'  cos  0 


:) 


(11) 


If  at  the  same  time  the  origin  be  changed  to  the  point  (/i,  k),  the  re- 
quired formidce  will  he 


x  —  x'  cos  0  —  y'  ^in  d  -\^h 
y  —  x'  ^\n  S  ^  y'  cos  d  -\-k 


:} 


(12) 


This  transformation  is  clearly  obtained  by  combining  the  two 
formulae  (10)  and  (11). 

Ex.  1.    Prove  by  means  of  (11) 

cos  (B  -|-  6^)  =  cos  6  cos  0^  —  sin  6  sin  ^^, 
sin  (9  +  ^0  =  sin  d  cos  e^  -j-  cos  0  sin  d\ 

Ex.  2.  Show  by  the  use  of  (12)  that  the  area  of  a  triangle  is  the  same 
function*  of  the  coordinates  of  its  vertices  referred  to  any  set  of  rectan- 
gular axes. 

To  turn  a  set  of  oblique  axes  through  an  angle  0  without  changing 
the  origin,  we  have,  since  w'  =  a>, 

X  =  \x'  sin  (d)  —  ^)  —  y'  sin  d~\  esc  w,  ^ 

.  M^  (13) 

y  =  [^x'  sin  0  -\-  y'  sin  (w  +  0)]  esc  (o.  J 

*  It  can  now  be  shown  that  formulae  proved  for  points  in  the  first  quadrant  will  hold 
for  points  in  any  quadrant.    (See  note  under  §  7.) 


96  CHANGE   OF    AXES.  [67. 

To  pass  from  rectangular  axes  to  oblique  without  changing  the  origin, 
we  have 

X  —  x'  cos  0  -\-  y'  cos  («>'-|-  0),^  . 

2/ =  a?' sin  ^ -f-/ sin  (w' -}-<?).  J  ^     ^ 

If,  in  making  this  transformation,  the  a:^-axis  is  not  changed, 
^  =0  and  the  required  formulae  are 


=  a?'+^'cos^',  1 

=  /sin.'.  }      •  (^') 


X  = 

y 

To  pass  from  oblique  to  rectangular  axes  having  the  same  origin,  we 
have 

X  =  [x'  sin  (o>  —  0)  —  y'  cos  (to  —  6)']  CSC  w, 

O). 


/ 1  f*  \ 

y  =  [x'  sin  0  -{-  y'  cos  ^]  csc  ' 


If  this  transformation  is  effected  without  changing  the  a;-axis, 
^  =  0  and  these  formulae  reduce  to 

X  =x' ?/COt  to.   ] 

,  \  (17) 

y=^y'  CSC  o).  J 

What  do  the  formulae  (13),  (14),  (15),  (16)  become  when  the 
origin  is  also  changed  to  the  point  (h,  k)l 

Observe  that  in  making  all  these  transformations  attention 
must  be  paid  to  the  signs  of  h,  k,  and  0. 

67.  The  degree  of  an  equation  can  not  be  altered  by  any  change  of 
the  axes. 

The  expressions  giving  x  and  y  in  terms  of  x'  and  y'  are  linear, 
that  is,  always 

x=lx'-\-my'-{-n,     y  ^=l'x' -\-m'y' -\-n' , 

where  either  /  or  m  is  not  zero,  and  also  either  V  or  m'. 
Any  term  in/(x,  y),  say  ax^y'^,  becomes 

a{lx'-\'  my'-^  nyiVx'Ar  m'y'^  n'y, 

and  contains  at  least  oue  term  of  degree  j9  -{-  q,  either 

alVx'P^^,     alm'x'Py'^,     aVmx'^y"^,     or     amm'y'^^^-, 

perhaps  other  terms  also,  but  none  of  higher  degree. 

Hence  the  degree  of  an  equation  can  not  be  raised  by  any  trans- 
formation of  coordinates.     Neither  can  it  be  lowered;  for  if  it 


69.]  CHANGE   OF   AXES.  97 

were,  by  changing  back  to  the  original  axes,  and  therefore  to  the 
original  equation,  the  degree  would  be  raised. 

Otherwise.  The  degree  of  an  equation  (and  also  of  its  locus)  is 
the  number  of  points  in  which  its  locus  is  cut  by  a  straight  line ; 
and  this  number  can  not  be  altered  by  any  transformation  of 
coordinates. 

Transformation  in  Polar  Coordinates. 

68.  To  turn  the  initial  line  through  an  angle  a  without  changing 
the  pole. 

LetangleXOX'^a. 

Let  P  be  any  point  whose  co- 
ordinates arc/o,  ^,  referred  to  OX, 
and  p,  d\  referred  to  OX'. 

Then  6  z=z  0'-\-  a,  while  p  is  not 
changed. 

Hence  the  desired  transforma- 
tion is  effected  by  simply  writing    o" 
^  4-  a  in  the  place  of  e.     [Of.  §  66,  (10).] 

69.  To  change  the  pole,  the  direction  of  the  initial  line  remaining 
the  same,  or  changed  by  an  angle  a. 

This  transformation  can  be  performed  by  first  changing  the 
given  equation  to  rectangular  coordinates  [§  6,  (2)]  ;  then  mov- 
ing the  origin  to  the  new  pole  [§  66,  (10)]  ;  then  transforming 
back  to  polar  coordinates  [§  6,  (1)]  ;  and  finally,  if  desired,  turn- 
ing the  initial  line  through  the  angle  «  (§  68). 

EXAMPLES. 
Transform  to  parallel  axes  through  the  point  (—  3,  2) 

1.  2/'  — 4^  +  42/ +  16=0. 

2.  2x^-\-3y^—12x-i-2y-\-29  =  0. 

What  are  the  equations  of  the  following  loci  when  referred  to  parallel 
axes  throughout  the  point  (a,  b)  ? 

3.  (x  —  ay-\-(y  —  by=r\ 

4.  xy  —  ax  —  hy-\-ah  =  a^. 

5.  2/2  — 262/  +  4ax=:4a2— 6^ 

6.  62(x2— 2ax)-f  a2(2/'  — 262/)  +  a262=0. 
8 


98  CHANGE   OF    AXES.  [69. 

Transform  by  turning  rectangular  axes  through  an  angle  of  45°. 

7.    x''  —  y^=a\  8.    lx'  —  2xy-{-7y-'=2. 

9.    2{y-\-x)  =  (iy  —  xy.  10.    ax''-{-2hxy -{-ay^  =  i. 

11.    a;*  +  6a;V  +  2/'  =  2.  12.    2xy(x'--^y'') +  1  =  0. 

13.  Transform  — [-  ^  =  l  by  turning  the  axes  through  tan-^  — . 

14.  What  does  2x^  —  3xy  —  2y-  =  5a^  become  when  the  axes  are  turned 
through  tan-i  2? 

15.  If  the  axes  be  turned  through  an  angle  of  30°,  what  does  the  equation 
9a;2  _  2\/Sxy  +  iiy^  =  4  become  ? 

16.  Show  that  the  equation 

2x'--{-xy-y'-\-3x  —  y-i-2  =  0 

can  be  reduced  to  2x'^  -]-xy  —  2/^  =  9,  by  transforming  to  parallel  axes 
through  a  properly  chosen  point. 

17.  The  equation  of  a  line  referred  to  axes  inclined  at  30°  isy  =  Sx  —  2. 
Show  that  its  equation  referred  to  axes  inclined  at  60°,  the  origin  and  x-axis 
not  being  changed,  is  (3  +  i/3)2/  =  3a;  —  2. 

18.  The  equation  of  a  curve  referred  to  axes  inclined  at  60°  is 
2^2 +  5x1/ +  2/^  =  2.  Find  its  equation  referred  to  rectangular  axes  such 
that  the  two  a;-aies  coincide. 

19.  Transform  y^  +  iay  cot  ,3  =  4ax  from  rectangular  to  oblique  axes 
meeting  at  an  angle  /3,  leaving  the  x-axis  and  origin  unchanged. 

29.  Show  that  the  formulae  for  transforming  from  rectangular  axes  OX, 
OY  to  oblique  axes  0X\  0Y\  such  that  the  angle  X'OY'=  w,  and  OX 
bisects  the  angle  X^OY\  are 

X  —  (2/^+  x^)  cot  ^w,    y  =  {y\ —  x')  sin  -Jw. 

21.  Apply  the  formulae  of  Ex.  20  to  the  equations 

-'±|'  =  1,    when6;  =  2tan-i-. 

22.  Prove  that  the  formulae  for  passing  from  axes  inclined  at  an  angle 
u>  to  axes  bisecting  the  angles  between  the  original  axes  are 

x  =  |(x''sec|w  —  2/' CSC  fw),    y  =  l{x^ sec  lo-\-y^ esc  ^w). 

Use  the  formulae  of  Ex.  22  and  thus  transform 

23.  x2+x2/  +  2/'  =  8,    when6;=.60°. 


2i,     \    ^    ^y  -r      f       when  o;  =  2  tan-i  ^. 

1  4xv  =  a^  +  62  J  *  a 


71.]  CHANGE    OF    AXES.  99 


Transformation  of  Functions  of  Two  Linear  Expressions. 

70.*    The  equatioDS  giving  the  values  of  x  and  y  in  terms  of 
x'  and  y'  [(8),  §  66]  may  be  written 

a?  sin  w  =  }>x'  +  !^y'  +  ^^ 


=  Ax/-\-fiy'-\-^,     ^ 
=  X'x'-^!/y'^>',  J 


(1) 

y  sm  (o  =z  A'x'-j-  ix'y' 

In  like  manner  we  should  find  the  equations  giving  x'  and  y'  in 
terms  of  x  and  y  to  be 

x'  sin  </>'=  Ix  +  my  -f  n,      ^ 

\  (2) 

2/'  sin  a>'==:  Vx  -\-  m'y  -f  ^',  J 

where  ^j;^  -|-  m?/  -f  n  =  0  (3) 

and  l'x-^m'y-^n'=0  (4) 

are  the  equations,  in  the  distance  form,  of  the  new  axes  0'  F'  and 
O'X',  respectively,  referred  to  the  old;  and  m'  is  the  angle  between 
O'X' and  O'F. 

If,  then,  the  given  equation  is  a  function  of  the  linear  expres- 
sions Ix  -\-  my  -f  n  and  Vx  +  m'y  -\-  n',  the  new  equation  is  ob- 
tained at  once  by  writing  x'  sin  w'  in  the  place  of  Ix  -j-  my  -\-  n, 
and  y'  sin  w'  in  the  place  of  Vx  -f  m'y  -\-n\ 

That  is,  if  the  given  equation  be 

f{lx  -\-  my  -(-  n,  Vx  +  m'y  -\-  n')  =  0,  (5) 

the  new  equation  referred  to  the  lines  (3)  and  (4)  will  be 

f(x'  sin  (o',  y'  sin  a;')  =  0 ;  (6) 

or,  if  the  new  axes  be  rectangular, 

f{x',y')=0.  (7) 

71.    Illustrative  Examples. 

Ex.  1.     What  ia  the  locus  of  the  equation  ' 

(2a;  — 2/  —  4)2  +  4(a;  +  2y  — 7)^  =  80?  (1) 

Dividing  by  5  gives 


100 


CHANGE   OF    AXES. 


[71, 


where  the  linear  expressions  within  the  parentheses  are  both  in  the  distance 
form. 


Take  a;  +  2i/  —  7  =  0  for  the  new  x-axis,  0'X\  and  2x  —  i/  —  4  =  0  for  the 
new2/-axis,  0^Y\ 
Then,  since  the  new  axes  are  rectangular, 


2a;  —  y  —  4 
75        ' 


y'- 


x  + 


T/5 


[(2),  §70] 


Writing  x^  in  the  place  of 


2x  — y 
1/5 


,  and  y^  in  the  place  of 


x  +  2y-l 


1/5 


in  (2)  gives  for  the  new  equation 


a;^^  +  4r^  =  16,    or    ^  +  ^  =  1, 
which  represents  an  ellipse  (§  34)  whose  semi-axes  are  4  and  2. 


(3) 


Ex.  2.     What  is  the  locus  of 

(3x-4^  +  12)2  =  5(4x  +  32/-h4)?  (1) 

Dividing  by  25  gives 

/3x-%-M2y _  /4xj-%_+4\ ^  ^2^ 

Take  3x  —  ^y-\-i2  =  0  for  the  new  x-axis,  O^X^,  and  4x  +  3?/  +  4  =  0  for 
the  new  2/-axis,  0^Y\ 
Then  since  angle  X'0'Y'  =  90°, 


._4x  +  %±i         ._3x-%-fl2 
X-         g         ,     y-  g  . 


[(2),  §70] 


Therefore  the  new  equation  referred  to  O^X^  and  O^Y^  is 

y''  =  x\  (3) 

Hence  the  locus  is  a  parabola  (§  37),  and  lies  on  the  positive  side  of  the 
line  4x  +  3i/  +  4  =  0. 


71.] 


CHANGE   OF   AXES. 


101 


Ex.  3.    Find  the  equation  of  the  locus  represented  by 
5(2x  - 1/  —  3)2  +  (3a;  —  iy  —  Sy=  45, 
when  the  new  x-axis  is  the  line 

'Sx  —  iy  —  S  =  0, 
and  the  new  y-axis  is  the  line 

2x  —  2/  —  3  =  0. 
Equation  (1)  may  be  written 

2x  —  y  —  ^W   ^a;  — 4y  — 8Y_9 


/2a;  — y  — 3y      /3a;  — 4y  — 8y_  9 
V       1/5       7"^V         5         )-h 


(1) 
(2) 
(3) 

(4) 


If  w^  be  the  angle  between  the  lines  (2)  and  (3),  then  from  equation  (5), 

§  48,  tan  w^=  ^,  and  therefore  sin  w^=  — ^. 

Let  p  and  q  be  the  perpendiculars  drawn  from  any  point  on  (4)  to  the 
lines  (3)  and  (2)  respectively  ;  then  from  (2)  §  70,  or  (5)  §  50, 


,  .      ,      a;'       2a;  — 2/  — 3 


q  =  y^  sin  0)^ 


y'  _3x  — 4y  — 8 
i/5~  5 

/n2        9 


•••  mHM-h  -  -+^-«. 


(5) 
(6) 
(7) 


is  the  required  equation. 
The  locus  is  enclosed  by  the  lines  x^=  i 3,  and  y^=  ±3;  construct  it. 
Observe  that  if  we  substitute  p  and  q  in  (4)  we  get 


P'  +  9'  =  §-, 


(8) 


102  CHANGE   OF   AXES.  [71. 

which  is  also  an  equation  of  the  locus  referred  to  the  same  new  axes,  but 
expressed  in. t^rms  of  perpendiculars  upon  the  axes  instead  of  parallels  to 
the  axes;  i.  e.  we  have  the  equation  in  a  new  system  of  coordinates. 

Ex.  4.    Find  the  ejjuation  of  the  straight  line 

9x  +  ly-\-U  =  0  (1) 

when  the  new  x-axis  is  the  line 

x  +  32/  +  6  =  0, 
and  the  new  y-axis  is  the  line 

Sx  —  y-^3  =  0. 

Assume  9x  +  7i/  +  14  =  l(3x  —  2/  +  3)  +  m(x  +  32/  +  6)  +  A; 

=  (3Z  +  m)a;+(3m  — l)y  +  (3J  +  6m  +  fc).         (2) 

Equating  coefficients  in  (2)  gives 

3Z  +  m  =  9,    3m  — i  =  7,    31 -{- 6m -]- k  =  U. 

Whence  1  =  2,    m  =  S,    k=z  —  10. 

Hence  equation  (1)  may  be  written 

2(3x-2/-3)  +  3(x  +  32/  +  6)-10=0,        "  (3) 

^Sx  —  y^3\   ,  o/x  +  3y  +  6\ 

H       /lO      j+^(       ,/10      j-viO  =  0.  (4) 

...    2x'-\-3y'=^10  (5) 

is  the  required  equation,  since  the  new  axes  are  rectangular. 

EXAMPLES. 

'  If  the  lines  x  —  y-\-l  =  0  and  x-\-y  =  2he  taken  as  new  axes,  what  are 
the  equations  of  the  lines 

I.  x  =  2,y  =  3?  2.    5x  +  2/  — 4  +  3^/2  =  0? 

3.    x—liy-\-m  =  0?  4.    ax-\-by  +  c  =  0? 

"When  referred  to  the  lines  3a;  —  41/  +  4  =  0  and  ix-\-3y  =  6  as  axes,  what 
are  the  equations  of  the  lines 

5.    18x4-^  =  4?  6.    x  —  18y  +  U  =  0? 

7.    8x  — 3l2/  =  20?  8.    22x  —  21y-\-6  =  0? 

9.    ^x  —  y  =  0?  10.    4x  +  52/  +  20  =  0? 

Find  the  equations  of  the  following  lines  when  the  lines  2x  —  y  =  4:  and 
x-\-2y  =  6  are  taken  as  axes : 

II.  2x—ny—12-\-y/6  =  0.  12.    7a;  — 63/  — 5  =  0. 
13.    i2x  —  y  =  22.  14.    x-\-i2y  =  l. 

Find  the  equations  of  the  following  straight  lines  in  oblique  coordinates, 

the  new  axes  being 

y  =  3x  +  6    and    3y  =  x^3: 


71.]  CHANGE  OF   AXES.  103 

15.    4y  — 4x  — 9  +  v/10  =  0.  16.    iix-\-3y  =  S. 

17.    xH-52/  +  5  =  0.  18.    ny  —  ilx  =  2Q. 

Find  the  equations  of  the  loci  represented  by  the  following  equations 
when  the  lines  represented  by  the  linear  expressions  which  they  contain 
are  chosen  for  the  new  axes  of  coordinates : 

19.  (4a;  +  32/-f-15)'^=5(3a;  — 4i/). 

20.  9(2x  — 32/  + 4)^4- 4(3a;  + 22/— 5)2=468. 

21.  (x  +  2/-4)'^  +  4(x-2/  +  2)  =  0. 

22.  3(a;  -\-Sy  —  4)^  —  4(3x  —  y-\-Qy=  120. 

23.  4(2x  — 4y  + 7)2  + 5r2a;  — 2/4-7)2=80. 

24.  4(5x  +  12^  +  24)2 —(12a;  —  5y-j-  15)2=  676. 

25.  (2/  — 3a; +  3)2=  20(32/  — X  — 6). 

26.  3(3a;  —  42/  —  12)2  _^  io(2a;  —  2/  +  4)2  =  150. 

27.  5(x  — 32/  — 4)2  +  4(a;  +  22/  +  2)2=200. 

28.  (2/  — 3x  +  3)2  — 2(2/  +  2a;  — 4)2=8. 

29.  2(x  +  y)=r{y-xf. 

30.  V2{y-xf={x-\-y  —  2)\- 

31.  (a;  +  22/  +  4)(2a;  — 2/)2=50i/5. 

32.  (x—y)2  +  2(a;-2/)(a;  +  2/+l)-(x  +  2/  + 1)^=2. 

33.  (Ix  +  m2/  +  n)(Vx  +  m'y  +  n')  =  0. 

34.  (2/-32  +  3)(2/  +  2a;-4)  =  25i/2. 

35.  Show  that  when  the  lines  2x  —  y-\-2  =  0  and  x-\-2y  =  0  are  the  axes 
of  coordinates  the  equation  of  the  locus  given  by 

2a-2  +  3x2/ +  232/2  +  2x  —  261/ +  13  =  0 

is  x2  —  3xy  +  42/2  +  2/5(a;  —  22/)  +  1  =  0. 

36.  Transform  the  equation 

9x2  —  16x2/  —  42/2  +  2x  —  91/  +  48  =  0 
to  the  oblique  axes  whose  equations  are 

y  —  3x  =  0    and    2/  +  2x+l  =  0. 

37.  Find  the  equations  of  the  old  axes  referred  to  the  new,  and  check 
the  results  obtained  in  numbers  35  and  36  by  passing  back  to  the  original 
equations. 


104 


PARAMETER   COORDINATES. 


[72. 


Parameters  of  Two  Loci  as  Coordinates  of  Points. 

72.*  The  point  (a,  b)  has  been  defined  as  the  point  for  which 
x  =  a  and  y  =  b',  i.  e.  the  point  of  intersection  of  the  two  lines 
whose  equations  are 

X  —  a  =  0     and     y  ^-b^O. 

Now  a  and  6  are  the  parameters  of  these  lines,  and  for  every 
pair  of  values  of  a  and  b  they  both  have  a  definite  position.  If 
a  and  b  vary  continuously  and  in  such  a  manner  that  b  =f{a)y 
these  lines  change  their  positions  simultaneously  and  continu- 
ously, and  their  common  point  will  describe  a  continuous*  curve 
whose  equation  is  y  =f(x). 

Likewise  in  polar  coordinates  the  point  (r,  a)  is  the  intersec- 
tion of  the  circle  and  the  straight  line 

p  =r     and     <?  =  a ; 

and  here  also  the  coordinates  of  the  point  are  the  parameters  of 
the  curves  whose  intersection  determines  the  position  of  the 
point.  If  these  parameters  vary  so  that  r=f{a)^  the  intersec- 
tion of  these  two  curves  will  move  so  that  p  =f(^0). 

Hence,  in  both  the  Cartesian  and  polar  systems,  the  coordinates 
of  a  point  may  be  regarded  as  the  parameters  of  two  loci  whose  in- 
tersection determines  the  position  of  the  point,  and  vice  versa. 

These  are  but  special  cases  of  the  following  general  principle : 


73.*     Let 

Fiix,y,a)  =  0         (1) 

and        F,(x,y,b)  =  0        (2) 

be  algebraic  equations  of  two 
curves,  a  and  b  being  arbi- 
trary parameters. 

If  particular  values  be  as- 
signed to  these  parameters, 
two  fixed  curves  A  and  B  will 
be  obtained  which  intersect  in 
P.     Now  if  a  varies  while  b 


*  See  second  note  under  §  73. 


73.]  PARAMETER   COORDINATES.  105 

remains  constant,  the  curve  A  will  change  its  position  while  thei 
curve  B  remains  fixed,  and  hence  F  will  move  along  B.  In  like 
manner,  if  h  varies  while  a  is  constant,  P  will  move  along  the  fixed 
curve  A.  If  all  possible  values  be  assigned  to  a  and  h  independ- 
ently, the  curves  A  and  B  move  independently  and  all  points  in 
the  plane  will  be  obtained,  provided  the  curves  A  and  B  both 
sweep  over  the  whole  plane.  Moreover,  to  each  pair  of  values  of 
a  and  h  there  corresponds  the  same  finite  number  of  points,  the 
number  depending  upon  the  degree  of  (1)  and  (2)  in  a;  and  i/; 
and  to  each  point  a  finite  number  of  values  of  a  and  h  depending 
upon  the  degree  of  (1)  and  (2)  in  a  and  6,  respectively.^ 

Hence  a  and  h  may  be  called  the  Parameter  Coordinates  of 
the  point  P. 

If  we  solve  (1)  and  (2)  for  a  and  h  respectively,  the  results 
may  be  written  in  the  form 

«  =  s^i(^,  2/).         ^=s^2(^,  ^),  (3) 

which  express  the  relations  between  parameter  and  Cartesian  co- 
ordinates. 

Suppose  the  parameters  a  and  ft  are  not  both  arbitrary,  but 
must  satisfy  the  equation 

6=/(a).  (4) 

Then  a  variation  in  a,  however  small,  causes  a  variation  in  h ; 
and  for  every  displacement  of  the  curve  A,  however  small,  there 
is  a  simultaneous  displacement  of  the  curve  B.  Hence  P  can 
take  only  a  fin^e  number  of  positions  on  any  single  curve ;  i.  e, 
P  can  not  move  to  all  places  in  the  plane.  If  we  assign  to  a  the 
particular  value  a^,  h  will  have  the  particular  value  h^,  and  we 
obtain  the  two  curves  A^  and  B^  which  intersect  in  Pj.  Likewise 
when  a  =  a2,  6  =  h^,  and  we  have  the  curves  Ao  and  B,  meeting 
in  Pa,  and  so  on. 

If  the  parameter  a  varies  continuously,!  then  will  h  also  vary 
continuously,  the  two- curves  A  and  B  will  be  displaced  in  a  contin- 
uous manner,  and  therefore  P  will  describe  a  continuous  curve  PQ. 

*  There  is  not  a  one  to  one  correspondence  between  a  pair  of  values  of  a  and  6,  and 
the  position  of  P,  unless  (1)  and  (2)  are  of  the  first  degree  in  x  and  y,  and  also  in  a  and  6. 

t  A  quantity  is  said  to  vary  continuously  from  one  value  p  to  another  value  q  when  it 
passes  through  all  values  intermediate  to  p  and  q  without  at  any  stage  making  a  sudden 
jump. 

It  is  here  assumed  that  6  is  a  continuous  function  of  a.  See  §  87;  also  Chrystal's  Al- 
gebra, Vol.  I,  Chap.  XV,  §  2  and  §  5. 


106  PARAMETER   COORDINATES.  [74. 

The  form  of  PQ  will  evidently  depend  upon  equation  (4),  which 
may  therefore  be  called  the  equation  of  the  locus  of  the  inter- 
section of  (1)  and  (2)  expressed  in  parameter  coordinates. 

74.*  To  find  the  locus  of  the  common  points  of  tivo  curves  whose 
equations  involve  one  and  the  same  independent  variable  parameter. 

Let  i^(^,  2/,  a)=0  (1) 

and  F,{x,  y,  b)  =0  (2) 

be  the  equations  of  two  loci  involving  the  variable  parameters  a 
and  b  which  are  connected  by  the  equation 

6  =/(«).  (3) 

Eliminating  b  from  (2)  by  means  of  (3)  gives 

Fr[x,y,f(a)l=0,  (4) 

and  we  have  the  equations  (1)  and  (4)  of  the  two  given  curves 
expressed  in  terms  of  the  same  arbitrary  parameter.  If  we  treat 
(1)  and  (4)  simultaneously  and  eliminate  a  we  obtain  an  equa- 
tion of  the  form 

<p(x,y)=0.  (5) 

Now  (1),  (4),  and  (5)  form  a  consistent  system  of  equations; 
i.  e.  all  values  of  x  and  y  which  satisfy  both  (1)  and  (4)  also  satisfy 
(5).  But  values  of  x  and  y  which  satisfy  both  (1)  and  (4)  are 
the  coordinates  of  the  common  points  of  their  loci.  Hence  the 
coordinates  of  all  points  common  to  the  two  given  curves  satisfy 
equation  (5)i  Moreover,  since  (5)  does  not  involve  a,  it  is  sat- 
isfied by  the  coordinates  of  all  points  common  t5  the  loci  of  (1) 
and  (4)  whatever  the  value  of  a  may  be;  i,  e.  by  the  coordinates 
of  all  points  on  the  curve  described  by  these  common  points  as  a 
varies. 

Therefore  (5)  is  the  equation  of  the  required  locus. 

Hence,  to  find  the  locus  of  the  common  points  of  two  curves  whose 
equations  involve  one  arbitrary  parameter,  treat  ihe  two  equations  simul- 
taneously and  eliminate  the  arbitrary  parameter. 

When  the  given  equations  contain  two  dependent  parameters, 
as  (1)  and  (2),  connected  by  an  equation,  such  as  (3) ,  the  required 
equation  (5)  is  found  by  eliminating  a  and  6  from  the  three  equa- 
tions (1),  (2;,  and  (3),  as  has  just  been  shown.     This  can  be 


75.] 


PARAMETER   COORDINATES. 


107 


accomplished  directly  by  substituting  equations  (3),  §  73,  in  equa- 
tion (3)  above. 

Hence  equations  (3),  §  73,  may  be  called  the  formulae  of 
transformation  for  changing  an  equation  from  parameter  to  Car- 
tesian coordinates. 

If  one  or  both  of  the  given  equations  contain  both  a  and  b,  the 
equations  can  each  be  expressed  in  terms  of  the  same  parameter ; 
or,  values  of  a  and  b  can  be  found  in  terms  of  x  and  y  as  before. 
Hence  the  locus  is  found  in  the  same  manner. 

Likewise  if  the  given  equations  contain  n  parameters  connected 
by  71  —  1  relations,  only  one  parameter  can  be  arbitrary ;  for  by 
means  of  the  n  —  1  equations  between  the  parameters  the  values 
of  all  can  be  found  in  terms  of  one,  and  the  given  equations  can 
then  be  expressed  in  terms  of  that  one. 

In  such  cases  the  locus  is  found  by  eliminating  the  n  parameters 
between  the  ti  +  1  given  equations.* 

A  few  examples  will  sufl&ce  to  make  this  general  theory  clear. 


Bi-PoLAR  Coordinates. 

75.*    Let  the  variable  curves  be  the  two  circles  whose  equations 
are  (§  32) 

(x-cy-\-f  =  r'  (1) 

and         ^  (x-{-cy-\-y'  =  r'\  (2) 

r  and  r'  being  the  variable  parameters. 


For  each  pair  of  values  of  r  and  r',  such  that  r  -j-  r'  >  2c  and 
\r  —  r'l  <  2c,  these  circles  intersect  in  two  real  points  P.     If  all 

*  a  fuller  discussion  of  this  subject  is  given  in  Cliap.  Ill,  Book  II,  of  Briot  and  Bouquet's 
Elements  of  Analytical  Geometry,  translated  by  J.  H.  Boyd. 


108  PAEAMETER   COORDINATES.  [76. 

positive  values,  which  satisfy  these  conditions,  be  assigned  to  r 
and  r'y  every  point  in  the  plane  will  be  obtained. 

The  variables  r  and  r',  which  represent  the  distances  of  the 
point  P  from  the  fixed  points  F  and  F\  are  called  the  Bi-Polar 
Coordinates  of  the  point  P. 

If  r  and  r'  vary  continuously  in  such  a  manner  that 

'•'=/W.  >  (3) 

then  the  circles  are  displaced  simultaneously  and  continuously 
and  their  common  points  describe  a  continuous  curve,  of  which 
(3)  is  the  bi-polar  equation. 
Solving  (1)  and  (2)  for  r  and  /,  respectively,  gives 


r  =  i/{x-  ey  +  f,         r'=  V {x  +  c)^  -j-  y\  (4) 

which  are  the  formulae  for  passing  from  bi-poIar  to  rectangular 
coordinates;  where  (c,  0)  and  ( — c,  0)  are  the  two  fixed  points 
of  the  bi-polar  system. 

EXAMPLES. 

Find  the  locus  of  P  in  rectangular  coordinates  when  its  bi-polar  coordi- 
nates satisfy  the  following  equations: 

1.  r  =  r\  3.    r  +  r^=2a.    (C/.  §34.) 

2.  r±r^=  2c.  4.    r  —  r'=2a.    (C/.  §  36.) 

5.  r'=nr.    (See  Ex.  24,  p.  50.) 

6.  rr^=c2.  -     Ans.    {x'  +  yy  =  2c\x' —  y"^). 

7.  Trace  rr^—  K  for  various  values  of  K^  taking  (±  1,  0)  for  poles. 

Using  the  points  (±  c,  0)  for  poles,  find  the  bi-polar  equations  of  the 
following  loci : 

8.  x^-\-y'^  =  0^.  10.    x  =  afX  =  Cfy  =  b. 

9.  x'-\-y^=aK  11.    2ix^  —  y^)  =  c\ 

12.  Show  that  the  formulae  for  changing  from  rectangular  to  bi-polar 
coordinates  are 

^=-4^'  y= -To 

13.  Show  that  the  bi-polar  equation  of  parabola  y^  =  Acx  may  be  written 


76.] 


PARAMETER   COORDINATES. 


109 


76  *    Let  the  two    given 
equations  be 

y=m(x  —  c)/       (1) 

and       y=m\x-\-c),        (2) 

7n  and  m'  being  arbitrarj^  pa- 
rameters. 

These  lines  pass  through 
the  fixed  points  F(c,  0)  and 
i^'( — c,  0),  respectively,  for 
all  values  of  m  and  m'.  When 
the  values  of  m  and  m'  are  given,  the  directions  of  these  lines  are 
determined  and  the  position  of  P  can  be  found. 

Hence  m  and  m'  are  coordinates  of  the  point  P. 

If  m  and  m'  may  take  any  values  independently,  P  will  move 
to  any  position  in  the  plane,  but  if  they  are  connected  by  the 
equation 

m'=Km),  (3) 

P  will  describe  a  definite  curve  (§73)  whose  form  depends  upon 
equation  (3).  The  equation  of  this  curve  in  rectangular  coor- 
dinates will  be  found  by  substituting  in  (3)  the  values 


m 


_    y 


y 


(4) 


X  —  c 
given  by  (1)  and  (2).     (See  §  74. ) 

EXAMPLES. 

1.  What  are  the  coordinates  of  O,  F,  F\  (0,  c),  (c,  c),  (—  c,  c),  (c,  0), 
( —  c,  0)  in  terms  of  m  and  m^? 

Find  the  locus  of  P  in  rectangular  coordinates  when 

2.  m''=  km.    Consider  the  special  case  fc  =  —  1. 

3.  mm^=  k.    Discuss  the  result  for  positive  and  negative  values  of  fc, 

especially  ±z  1 . 


4.  m  -f  m^=  k. 

6.  m{a  —  c)  =  m'(a  +  c). 

8.  2mm^  =  a  (m  -f  m-') . 

10.  ^FPF'=a, 


5.  m'  —  m  =  k. 

7.  2cmm^=6(m  —  m^). 

9.  m^m^'^  =m^  —  m^^. 

11.  ZXFP-\-ZXF'P=a. 


Consider  both  acute  and  obtuse  values  of  the  constant  angle  «. 


110 


PARAMETER   COORDINATES. 


[77. 


Let  angle  XFP  =  r  =  tan"^  m,  and  angle  XFP  =  d  =  tan"'  m\ 
Then  y,  d  are  also  coordinates  of  P,  suQh  that  for  every  pair  of 

values  of  y  and  d  there  is  one  and  only  one  position  of  P,  except  for 

points  on  the  line  FF'.^ 


77.*    Example. 


Let  OX  and  OYbe  two  fixed  lines 
and  Pi(iC] ,  2/i)  a  fixed  point.  Through 
Pi  draw  a  fixed  secant  P^BA  {meeting 
OX  in  A,  OY  in  B),  and  the  variable 
secant  PiDC  {meeting  OX  in  C,  OYin 
D);  also  draw  the  lines  AD  and  BC 
meeting  in  P.    Find  the  locus  of  P. 

Take  the  lines  OX  and  OF  as  axes 
of  coordinates. 
The    equation    of   PiBA  may  be 


^\       written  (§  60) 


y  —  yi=mi{x  —  xi), 
where  mi  is  constant. 
Similarly  the  equation  of  the  variable  secant  PiDC  is 

2/  — 2/1  =  m{x  —  xi), 

where  m  is  the  variable  parameter. 
Putting  x  =  0,  and  i/  =  0  in  (1)  and  (2)  we  find 


(1) 


(2) 


OA  =  x^-^, 
mi 


OC  =  xi  — 


m» 


OB  =  yi  —  mixif 


OD  =  yi  —mxi. 


Hence  the  equations  of  AD  and  BC  are  (§  41  and  §  60) 
X        ,         y 


Xi  — 


h       yi  —  mxi 


and 


mi 


-  + 


y 


Xi  — 


y\     2/1  —  wna^i 

m 


(3) 
(4) 


Equations  (3)  and  (4)  contain  one  and  the  same  variable  parameter,  m ; 
hence  the  locus  of  P  is  found  by  eliminating  m  between  these  two  equa- 
tions.   (§  74.) 

*As  a  result  of  this  general  theory  we  may  say  that,  in  the  most  general  sense,  the 
point  (a,  h)  is  the  intersection  of  two  curves  whose  equations  involve  a  and  6  as  arbitrary 
parameters.  When  the  two  curves  are  straight  lines  parallel  to  two  fixed  lines  we  have 
Cartesian  coordinates;  when  one  is  a  circle  with  a  fixed  centre  and  the  other  a  straight 
line  through  its  centre,  we  have  polar  coordinates;  when  both  curves  are  circles  having 
fixed  centres,  we  have  bi-polar  coordinates,  etc. 


77.]  PARAMETER   COORDINATES.  Ill 

By  subtracting  (4)  from  (3)  we  obtain 

^\ yi  —  mxi  ""i/i  — mix,/  '^^\yi  —  mxi      yi  —  mixj  ~    * 

which  simplified  gives 

yix-\-xiy  =  0.  (6) 

Therefore  the  locus  is  a  straight  line  passing  through  O. 


EXAMPLES. 

1.  A  trapezoid  is  formed  by  drawing  a  line  parallel  to  the  base  of  a  tri- 
angle.   Find  the  locus  of  the  intersection  of  its  diagonals. 

2.  Through  a  fixed  point  O  a  variable  secant  is  drawn  meeting  two  fixed 
parallel  lines  in  R  and  Q;  through  R  and  Q  straight  lines  are  drawn  in  fixed 
directions,  meeting  in  P.    Find  the  locus  of  P. 

3.  The  hypotenuse  of  a  given  right  triangle  slides  between  the  axes  of 
coordinates,  its  ends  always  touching  the  axes.  Find  the  locus  of  the  ver- 
tex of  the  right  angle. 

4.  Find  the  locus  of  the  intersection  of  the  lines 

'L  +  t  =  i    and    f  +  |  =  l, 
am  lb' 

where  I  and  m  are  variable  parameters,  such  that 

a-\-m  =  b-{-l. 

5.  Find  the  locus  of  the  centres  of  all  rectangles  which  may  be  inscribed 
in  a"  given  triangle. 

6.  A  variable  quadrilateral  is  inscribed  in  a  given  rectangle  so  that  its 
diagonals  are  perpendicular  to  each  other  and  parallel  to  the  sides  of  the 
rectangle.     Find  the  locus  of  the  intersection  of  its  opposite  sides. 

7.  Find  the  equation  of  the  locus  of  a  point  at  which  two  given  por- 
tions of  the  same  straight  line  subtend  equal  angles. 

8.  Find  the  locus  of  the  intersection  of  the  two  lines 

y  —  mx  =  aV'i-{-m^    and    my -\- x  =a  V 1 -\- m^ 
for  all  values  of  m. 


CHAPTER  V. 


SLOPE,  TANGENTS  AND  NORMALS. 


78.  Definitions.  Let  two  points  P  and  Q  be  taken  on  any 
curve  PQR,  and  let  the  point  Q  move  along  the  curve  nearer  and 
nearer  to  P;  the  limiting  position,  TT'j 
of  the  secant  PQ  when  the  point  Q  moves 
up  to  and  uUimatelij  coincides  with  P  is 
called  the  Tangent  *  to  the  curve  at  the 
point  P. 

The  straight  line  Pi\^  through  the  point 

P,  perpendicular  to  the  tangent  TT',  is 

called  the  Normal  to  the  curve  at  the 

point  P. 

The  Slope,  or  Gradient,  of  a  curve  at  any  point  is  the  slope 

of  the  straight  line  tangent  to  the  curve  at  that  point. 


79.      To  find  the  slope  of  a  curve  at  any  point. -\ 


Let  P{x,  y)  and  Q{x  -{- ^x^  y  -\-  %)  be  two  points  close  together 
on  any  curve  AB;  then  dx  is  the  difference  of  the  abscissas,  dy  the 
difference  of  the  ordinates  of  P  and  Q. 

*  This  definition  was  first  suggested  by  Roberval  (1602-1675) ,  but  was  stated  more  con- 
cisely by  Fermat  and  Des  Cartes.    (History  of  Math.— Cajori,  p.  173;  Ball,  p.  243.) 
t  Read  Ex.  1,  §  81,  in  connection  with  this  general  demonstration. 


79.]  SLOPE,  TANGENTS   AND   NORMALS.  113 

Let  the  secant  PQ  meet  the  a^-axis  in  S,  and  let  the  tangent  line 
at  P  meet  the  ic-axis  in  T. 

Draw  the  ordinates  MP,  NQ,  and  draw  PR  parallel  to  the  a?-axis. 

Then  PR  =  dx,     BQ==  hj. 

Let  the  equation  of  the  curve  be 

2/=/(^).  (1) 

Then  at  the  points  P  and  Q  we  have 

OM=x,  MP  =  y=J{x), 

ON=x  ^dx,     Nq  =  y  ^  ^y  =  f(x  +  ^x). 
.-.      dy=f(x-^dx)—fiix). 

Also  '    ta.nXSQ  =  i.a.nEPQ  =  ^=^. 

JrjLi  ox 

•••    tanX.g  =  g=/(-  +  ^-)-/(-).  (2) 

The  slope  of  the  tangent  TP,  which  is  the  slope  of  the  curve  at 
the  point  P,  is  the  ultimate  slope  of  the  secant  SPQ  when  the 
point  Q  moves  along  the  curve  close  up  to  P;  i.  e. 

dy 
tan  XTP  =  lim  tan  XSQ  =  lim  ^  as  §  approaches  P. 

ox 

When  the  point  Q  approaches  the  position  of  P  as  a  limit,  the 
differences  dx  and  dy  simultaneously  approach  zero  as  a  limit,  and 

the  limiting  value  of  the  ratio  -y-  is  denoted  by  -^ ;  therefore  in  the 
limit  we  have 

tan  XTP  =  g  ^^H^;/(^  +  ^^)-/(-r):  ^33 

The  ratio  represented  by  the  last  member  of  equation  (3)  is 
also  a  function  of  x;  and  if,  x  being  regarded  as  fixed,  this  ratio 
has  a  definite  limiting  value  as  dx  becomes  zero,  this  limiting 
value  is  called  the  Derived  Function,  or  the  Derivative  of 
f{x)  with  respect  to  x,  and  will  be  denoted  hy  f(x)  ;  *l.  e.  if 

y  =  fix),  then  ^J'-=nx). 

♦The  sign  "  =  "  in  these  conditions  for  a  limit  (Jaj  =  0)  is  to  be  understood  to  mean 
becomes  equal. 

9 


114  SLOPE,  TANGENTS   AND   NORMALS.  [80. 

Hence  to  find  the  slope  at  any  point  of  a  curve  whose  equation 
is  in  the  form  y  =f(x)  we  find /(a;),  the  derivative  of  f(x)  with 
respect  to  x,  and  in  this  substitute  the  abscissa  of  the  given  point. 

To  find  the  derivative  of  a  function  of  x,  denoted  hy  f(x)j  we 
assign  a  small  increment  dx  to  x,  producing  an  increment,  de- 
noted by  f(x  +  dx)  — f(oc),  in  the  function,  and  then  find  the 
limiting  value  of  the  ratio 

Kx-^dx)—f(x) 
dx 
as  3x  vanishes. 

E.g.f  let  fix)  =  ar>,  then 

lim    f{x-\-6x)-f{x)  _    lim    (x-\-<^x)'-a^ 
rix)  =  dx  =  0 j^ -  ^x  =  0 ^ — 

=  ^l^^Q  (5x4  _^  10x3  Jx  +  lOx^f^x-^  +  bxSar^  +  6x')  =  5xK 

The  operation  of  finding  the  derivative  of  a  function  is  called 

differentiation. 

It  is  to  be  carefully  noticed  that  in  the  definition  of  a  deri .  a- 

,    dy 
tive  given  above  we  speak  of  the  limiting  value  of  the  ratio  ~-,  and 

^x 

not  of  the  ratio  of  the  limiting  values  of  dy  and  dx.     The  latter  ratio 

"is  indeterminate,  on  the  face  of  it,  being  of  the  form  — .     To  give 

the  latter  definiteness  we  now  define  it  as  equal  to  the  former. 
That  is,  the  ratio  of  the  vanishing  increments  of  function  and 
variable  is  the  limit  that  the  ratio  of  their  finite  increments  ap- 
proaches when  these  finite  increments  at  last  vanish. 

80.     Examples  of  limiting  values  of  ratios  of  vanishing  quantities. 

(1.)    Let  K  be  the  area  of  a  square  whose  side  is  x. 
r  lim  area  ~|  _  0 

L  lim  side  J  X  =  0~  0* 

^  ^  lim   K       lim   x^        lim    .  ,      ^ 

But  ^^0^-=a;  =  0^=x  =  0(^)  =  0- 

(2.)  Let  I^  be  the  area  of  a  rectangle  with  a  constant  base  b  and  a  vari- 
able altitude  x. 

riim^n  0 

Then  i-     ^  a  =  a- 

L  lim  X  J  X  =  0       0 

^   ,  lim   K        lim    bx      . 

But  ^^Q~  =  ^^Q~^  =  b. 


80.]  SLOPE,  TANGENTS   AND   NORMALS.    '  115 

(3.)  Let  V  be  the  volume,  T  the  total  surface,  C  the  circumference  of  the 
base  of  a  right  circular  cylinder  whose  altitude  is  constant  and  radius 
variable. 

r^^"L^l        -9.       riimr~i        _o 

^'^^^  LlimOjr  =  0~0'         LlimFjr  =  0~0' 

_   ,                lim   T       Urn  2:rr(r  +  /i)        lim   ,     .   .,      , 
But  y^Oe^^^O 2^ =  r--=0('*  +  '^)="^ 

lim    T        lim   27rr(r-\-h)        lim   2(r4-h)       2h 
and  r  =  Ov=  r  =  0       ^^h        =  r  r^Q—fh~  = -W^^"^ ' 

If  S  be  the  convex  surface,  find  ^  _  q  -=. 

^  ^^  Liim(a:2— a2)Jar  =  a      0' 

,  lim   (a:  —  af  _   lim    a;  — a  _  ^ 

a:  =  a  ^nr^  ~x  =  aa;4-a  ~ 

(5.)    Find  ^ifo-"'"^"^ 


Multiplying  both  numerator  and  denominator  by  1  +  V^l  —  «^  gives 
lim   1  —  1^1  —x^  _    lim  a:-  _    Hm  i  _  1 

EXAMPLES. 
Find  the  limits  indicated  in  the  following  expressions : 
.^C^    lira   x^  —  a^  ^(-^  lim   x*  —  a* 

lim  (x  —  a)3  ^-.  lim    Sx^  _  6x  +  3 


:«? 


lim  (X  —  ay  ^-\  iim    ijx'  —  Dx-t-i5 

a^  =  ax=»-ax2  — a^x  +  a-^*  fiX x  =  1  2x2  _  4^; 4_ 2 


,^57     lim  a;'^  ^^    lim     gx^  +  x  — 1 


iP 


x:-0a_v/a2^=^2'  (^x=QO   a;2_a._|_2  ' 

^„)    lim    1/4  +  x  — /4— X  O    lim       .-  ,  ,     . 


6? 


^  1     lim   sin  x      .  /^rCpi    lim     sec  x      , 


&? 


^  =  Otanx-^- 

lim   1  —  cos  X 
a^^O     sin^x 

J. 

lim   sin  x         1 

im   1 
=  0 

ban 

a: 

X=:0       a;                X 

X 

lim   tan  x  —  sin  x 


13.    a."l"o^^  =x'=0^^^  =  l• 
14.    If  V  be  the  volume,  T  the  total  surface,  8  the  convex  surface,  C  the 
circumference  of  the  base  of  a  cone  of  revolution  whose  altitude  h  is  con- 
stant, show  that 

lim    r  _  1       lim   T  _  lim    T  _  .        lim    T  _  „ 

r  =  0(7— 2'    r  =  Op^~°^'        r  =  05~^»    t  —  ^^~'^* 


116 


SLOPE,  TANGENTS   AND   NORMALS. 


[81. 


81.     Examples  of  derivatives  and  slope  of  curves. 
Ex.  1.    Find  the  slope  of  the  curve  whose  equation  is 


(1) 


Let  P(x,  y)  and  Q{x  +  '^a;,  y  +  (^y)  be  any 
two  points  close  together  on  the  curve ;  and 
let  TP  be  the  tangent  at  P. 

Then  at  P,       2/  =  a:^  -f-  a,  (2) 

and  at  Q,      y -[- dy  =  x -\- ^x)^ -\- a.  (3) 

Whence 

{y  +  ^y)  —  y^  (^  +  ^xf  -\-a  —  {x'-\-a) 

6x  ^x 

=  tan  RPQ.  (4) 

.     ^y 


^x 


=  2x  +  c^x  =  tan  XSQ. 


(5) 


When  Q  coincides  with  P,  or  as  we  say, 
X   proceeding  to  the  limit  (h  =  0,  we  have  (§79) 

^  =  2x  =  tan  XTP.  (6) 

dx 

Hence  the  slope  of  the  curve  at  any  point  is  equal  to  twice  the  abscissa 

of  the  point. 

At  Po,    x  =  0. 

.  • .    PoTo  is  parallel  to  the  x-axis. 

At  Pi,    ic  —  J, 

.-.    tanJ^TiPi^l. 

A.t  i2>       •'^  "~  2J 

.-.    tanXr2P2  =  3. 

At  Pz,    x  =  —  ^, 

.-.    tanXr3P3  =  — 1. 


E^    Pi 


Find  the  slope  of  the  curve  y  = 


We  now  have 


1 


Whence 


x-\-  ^x 
1 


(^X 


x{x-{-  (^X)' 


=f{x-j-6x)-fix)  = 

^y ^ 

6x  x{x-{-Sxy 

.      dy  ^    lim  / L__V_1. 

dx       6x  —  0\     x{x-\-^'X')/  x^' 

That  is,  the  slope  is  always  negative  and  varies  inversely  as  the  square 
of  the  abscissa  of  the  point. 


81.]  SLOPE,  TANGENTS   AND   NORMALS.  117 

BiTs)    Let  y  =  \/x  he  the  given  curve. 
Then  6y=Vx-{-^x  —  ^x 


Vx-\-6x-^x/^* 
6y  _                1 
and  —r~  —    /  • 

Sx         y^x-^6x-\-^X 

.     ^  ^    lim  1  ^  _J_ 

••     dx       <^''x  =  0^x  +  6x  +  i/x      2v/x* 

Verify  the  results  found  in  Exs.  2  and  3  by  constructing  the  loci. 

EXAMPLES. 

Find  the  slope  at  the  points  where  x  =  0,  dr  1,  ±  2,  etc.,  of  the  curves 
whose  equations  are 

1.  y  =  x^.  2.    y  =  X*. 

3.  y  =  ^,.  4.    2/^  =  x3. 

5.  2/  =  x3  — 4x.  6.    2/ =x*  — 20x2-1-64. 

7.  Find  the  slope  ofy=  Va^  -\-  x*,  where  x  =  0,  i  a,  oo  . 

8.  Find  the  slope  oiy~  V a? — x^,  where  x  =  0,  ±  a,  =h  ^a. 

9.  Find  the  slope  of   iOy  =  x'  —  3x  —  20,  where  x  =  0,   ±1,   =b  4. 

[Ex.  1,  §  22.] 
10.    Find  the  slope  oty  =  x  and  y  =  mx  +  b. 


Find  the  derivatives  of  the  functions 

11.    y  =  ^.  12.    2/  =  -^^-L_ 

^      x  +  a  ^      x^  —  a^ 

^^'    y  =  i^'  ^^'    2/  =  a^^  +  bx  +  c. 

15.    2^  =  sinx.     [See  Ex.  1,  §  104.]     16.    y  =  cosx. 

17.-    If,=_^4showthatf  =  -^, 

18.*    If  y  =f(x)  ■  ■Hx),  show  that  ^  =/(a!)-#'((r)  +  «*(x)-/'(x). 

19.-    Ifj,  =  ^,showthat|^  =  tW^^^(^^=;^5m^. 
^(x)'  dx  {.Kx)Y 

20.*    If  2/  =[/(x)]«,  show  that  -^  -  n[/(x)]'-'  -/^(x).    [Use  §  82.] 

21.*    Find  the  derivatives  of  the  functions 

,              ^                             .                         „      x  +  a  ax  —  b 
y  =  tan  x,  cot  x,  sec  x,  cscx,  sin  x,  cos  x,  cos^  x,  — ' —  , r-^. 


118  SLOPE,  TANGENTS   AND   NORMALS.  [82. 

82.     To  prove  that  for  all  rational  values  of  n 
lim    0^" —  a" 


X  =a 


X  —  a 


I.     When  71  is  a  positive  integer,  we  have 
lim  x*"  —  a"        lim 


X,  =  a"^^  =  x'=a (*""'  +  ^'"<'  +  *"'V  +  .  .  +  xa-^  +  a-') 


a"   ^  -|-  a"   ^  +  .   .   .  to  n  terms 


II.     When  n  = -,  a  positive  fraction,  we  put 
x=^if     and     a  =  6^ ; 
then     x^=^y^,     a^  =  b^,     and  when  ?/ =  6,     ir=a. 

lim    ic"  —  a" lim    x"^ — a^ lim    if  —  b^' 

'  '     x=a  x  —  a  ~x=a    x  —  a    ~~  V  =^  if  —  ¥ 

lim     y  —  b        pb^~^  .^ 

-y=.bf^=^^  (Case  I.) 

3  q 


III.     When  n  =  —  m,  we  have 

lim    x"  —  a" lim    x~'^  —  a~"* lim 

^  =  «  x—a  ~~x  =  a      x  —  a       ^ x  = 

1  „    , 


/  1      ^x"^  —  a^'X 

«\       x^a"^      x  —  a  ) 


whether  m  is  an  integer  or  a  fraction..    (Cases  I  and  II. ) 

Ex.  1.    Show  that 

limit       [/(x)]'»— [^(a;)]~         ^^,  ^-,     , 

Ex.  2.    Show  that 

lim     sin^  x  —  cos^  x  _  3 
x  =  45°sinx  —  cosx~2' 


83.]  slope,  tangents  and  normals.  119 

General  Rules  for  Differentiation. 

83.     Differentiation  of  an  integral  algebraic  function  of  one  variable 
with  rational  exponents. 

The  most  general  form  of  such  a  function  is 

ax**  +  bx**'^  -\-  cx"~^  +  .   .   .   -\-  kx  -\-l. 

Let     y  =f(x)  =  ax''  +  ft^""'  +  ex""-'  -{-...   -^  kx  +  I       (1) 

If  we  let  dx  =  h  =  (x  -]-  h)  —  x,  for  convenience,  then  will 

'>y=f(x  +  h)-f(x);  [§79] 

that  is, 

dy  =  al(x-\-hy  —  x"]  +b\_(x-{-hy-'—x''-'] 

-\-cllix  +  hy-'-x-'-]-j-  .   .   .   -{-k[(x  +  h)—x-].     (2) 

dy  _   r(x-\-hy—x''Xi  .r(x^hy-'—x"-' 


dx 


nx  +  hy-xn      r(x-^hy-^-x-n 

L(:x-\-h)—xV   I   (x-]-h)—x  J 


Proceeding  to  the  limit  dx=h  =  0,we  obtain,  by  applying  §  82 
to  each  term  of  the  second  member  of  (3), 

^  =  nax''-'+  b(n  —  l)x''-'-\-  c(n  —  2)x*'-'^  .   .   k=f{x).    (4) 

Hence,  if  f(x)  is  an  integral  algebraic  function,  we  find  f(x)  by 
multiplying  the  coefficient  of  each  term  by  the  exponent  of  x  in  that  term 
and  diminishing  each  exponent  by  unity. 

E.  g.,  if /(x)  =a:*— 2ar»+  3x2  _^  a;  — 4a;°  — 6x-i  -|-2a;-2_3a;-3, 

/''(x)  =  4«3_6a;2 -f  6x+ 1 +6a;-2  —  4a;-' +  9a;-*. 

Observe  from  (4)  and  (1)  that  the  derivative  of  the  sum  of  a  number  of 
terms  is  the  sum  of  the  derivatives  of  the  separate  terms,  and  also  that  the 
derivative  of  a  constant  term  is  zero. 

Ex.  1.    If  fix)  =f{x)  +f2{x),  show  that/^(a;)  =f/(ix)  +fAx). 

Ex.  2.  Show  by  constructing  a  figure  that  the  slope  of  y  =fi{x)  -i-fAx) 
is  the  sum  of  the  separate  slopes  of 

2/=/i(«)    and    y=f2(x). 


120  SLOPE,  TANGENTS   AND   NORMALS.  [84. 

84.     To  find  the  derivative  of  a  Junction  of  the  type  F(^x,  y)  =0. 

When  we  desire  to  differentiate  a  function  of  the  type 
F{x,  y)  =  0,  we  may  try  first  to  solve  the  equation  with  respect 
to  y,  so  as  to  put  it  in  the  form  y  =/(^) ;  or  to  solve  with  re- 
spect to  X,  so  as  to  bring  it  to  the  form  x  =fi(y)'  It  is  useful, 
however,  to  have  a  rule  to  meet  cases  when  this  process  would 
be  inconvenient  or  impracticable.  It  will  be  sufficient  for  the 
purpose  of  this  book  to  illustrate  the  rule  by  considering  the  gen- 
eral equation  of  the  second  degree  (§  53). 

Let     F(x,y-)=ax'-^2hxy^by'-\-2gx-h2fy-i-e=0.         (1) 

Let  P(^x,  y)  and  Q(x  -}-  ^x,  y  -\-  ^y)  be  two  points  close  to- 
gether on  the  locus  of  (1)  ;  then  at  P  and  §,  respectively, 

ax'  +  2hxy  +  hf  +  2gx  +  2/./  +  c  =  0,  (2) 

a(ix  +  dxy  +  2h{x  +  dx){y  +  5y)+h{y  +  dyf 

+  2g{x  +  <^^)-f  2/(^  J^8y)Jrc=0.      (3) 
Subtracting  (2)  from  (3)  gives 
a(2xdx  4-  dx")  +  2h{yr)x  +  x^y  -\-  dxdy) 

+  b(:2ydy  +  df)  +  2g3x  +  2fdy  =  0.      (4) 


^  ___  2ax  +  2hy  +  2gr  -f  gdx  +  2hdy 
Whence     -^— —     ~2ho^^\^hy  +  '2/  +  hdy       ' 

Proceeding  t3  the  limit,  when  8x  =^dy  =  0,  we  have 
dy  _       ax  -\-  hy  +  g 


(5) 


(6) 


dx  hx  -\-  by  -{-f 

Now  apply  to  ( 1 )  the  rule  deduced  in  §  83  and  differentiate 
first  with  respect  to  x  regarding  y  as  comtant;  then  differentiate 
with  respect  to  y  regarding  x  as  constant.  Denoting  these  partial 
derivatives  respectively  by  FJ(x,  y)  and  F/(x,  i/),  we  thus 

obtain 

F^'(x,y)  =  2(ax  +  hy-i-g)  (7) 

and  F;(x,y)=2{hx-i-by^f).      .  (8) 

.      djj  ^        FJ(x,  y)  ^       ax-\-  hy  -f-  g 
"     dx  F;(x,y)  hx-hby+f  ^   { 

which  expresses  the  rule  for  differentiating  any  function  of  the 

tjpeF(ix,  y)  =  0.  ^ 


85.]  SLOPE,  TANGENTS  AND   NORMALS,  121 


Tangents  and  Normals. 

85.      To  fnd  the  equations  of  the  tangent,  and  the  normal  at  any 
point  (x'y  y')  of  a  curve. 

dii' 
For  the  tangent,  m  =  y ,.  (§  79. ) 

dx' 
For  the  normal,  m  = —  ,-,.  (§  78  and  §  48.) 

Since  both  lines  pass  through  the  point  {x',  ?/'),  the  equation 
of  the  tangent  is  (§  49) 

^  — 2/-^.(^  — ^');  (1; 

and  the  equation  of  the  normal  is 

y  —  y'=  —  ^,(.^—^'')'  (2; 

dii' 
The  primes  in  -^^  denote  that  the  coordinates  x',  y'  cf  the  point 

of  contact  are  to  be  substituted  in  the  derivative  of  the  equation. 

Cor.     If  the  axes  are  oblique, 

dy  sin  y  , «  ^  ^  . 

-T-=^^ — ^=m.  (§59.) 

dx       sin  (w  —  y)^  , 

Hence  equation  ( 1 )  holds  also  for  oblique  axes.  * 

»  Examples  on  Chapter  V. 

Find  the  equations  of  the  tangent  and  normal  to  each  of  the  following 
curves  at  the  point  (x^,  y') : 

1.    y  =  x^.  Ans.    —y  —  --,  =  1. 

'^  x'       y' 


Ans. 

2y        X 

y'       X' 

Ans. 

3a:         y        . 

2x'      2y' 

Ans. 

3x        2y  _^ 

X'        y' 

Ans. 

Ans. 

xx^-{-yy^=l. 

*  The  theory  of  this  chapter  proves  what  has  hitherto  been  assumed  (see  note  on  logic 
of  plotting,  §  21),  viz.,  that  loci  of  equations  are  usually  smooth  curves  without  sudden 
changes  in  slope  or  curvature.  For,  since  the  slope  of  a  curve  f(x,  y)  =0  a.t  any  point 
(x,  y)  is  a  function  of  x  and  y,  a  small  change  in  x  and  y  will  ordinarily  produce  only  a 
small  change  in  the  slope. 


122  SLOPE;  TANGENTS   AND   NORMALS.  [85. 

7.  x2  — 2/2  =  1. 

8.  x^  +  y^  =  i.  Ans.     xx'""  +  yy'^  =  i. 

11.  What  are  the  equations  of  the  tangents  to  6,  7,  8,  9,  10  at  the  point 
(1,  0);  and  to  6,  9,  10  at  the  point  CO,  1)  ? 

Find  the  equation  of  the  tangent  to 

12.  2/*  =  ix  —  3x\  at  the  point  (1,  1). 

13.  iOy  =  (x-\-  1)2  at  the  point  where  x  =  9.     (Ex.  10,  p.  28.) 

14.  4(x+l)  =  (2/  — 2)- atthepointwherex  =  3.     (Ex.  10,  p.  28.) 

15.  {X  —  8)2  +  (2/  —  2)2  =  25  at  the  point  where  x  =  4. 

16.  x(a;2  -j-  2/^)  =  a[x^ —  2/0  ^^  *^®  point  where  x  =  0,  and  ±  a.     (§  38.) 

17.  Find  the  equation  of  the  tangent  to  f  —  j  +(-?)  =2,  and  show  that 
at  the  point  (a,  b)  it  is  the  same  for  all  values  of  n. 

18.  Show  that  the  curve  x^-\-y^  =  a^  becomes  steeper  as  it  approaches 
the  2/-axis,  and  is  tangent  to  the  axes  at  the  points  (d=  a,  0)  and  (0,  zt  a). 

19.  Let  y  —f{x)  and  y  =  F{x)  be  two  curves  intersecting  in  the  point 
',xi,  2/i),  and  let  (p  be  the  angle  at  which  they  intersect.    Show  that 

*^^^  =  -iT/^(^i^^(^)- 

What  is  the  condition  that  the  two  curves  shall  meet  at  right  angles?  , 
be  tangent  to  each  other? 

[The  angle  at  which  two  curves  intersect  is  the  angle  between  their  tan- 
gents at  the  point  of  intersection  of  the  curves.] 

20.  -  Find  the  angle  of  intersection  between  the  parabolas 

2/2  =  4ax    and    x"^  =  iay. 

21.  Show  that  the  confocal  parabolas 

2/^  =  2a(x  +  a)    and    y^  =  —  26(x  —  b) 
intersect  at  right  angles. 

22.  At  what  angle  do  the  rectangular  hyperbolas 

3.2  —  y2  —  (j2    ^^^    xy  =  b 

intersect?    Draw  several  sets  of  these  curves  by  assigning  different  values 
to  a  and  b.  ,  . 


85.] 


SLOPE,  TANGENTS   AND    NORMALS. 


123 


23.  Find  the  angle  at  which  the  circle  x^  +  2/^  +  2a;  =  12  intersects  the 
parabola  y^  =  9x. 

24.  Find  the  angle  of  intersection  between 

x^-\-y^  =  26    and    4.y'  =  9x. 

25.  Let  the  tangent  and  the  normal  at  any  point  P(x,  y)  of  a  curve  meet 
the  X-axis  in  Tand  N  respectively,  and  let  M  be  the  foot  of  the  ordinate  of  P. 


Prove,  by  the  use  of  equations  (1)  and  (2),  §  85,  the  following  formulae: 

dx 


Subtangentf     TM=  y 


dy' 


Subnormal,    MN=y 


_..  ^y 


dx 


Tangent,     TP  =  y^^l -\- (^J . 

Ncynnal,    PN=y^ij^{^y\\ 
dx 


Intercept,     OT  =x — y 


dy' 
dy 


Intercept,     OQ  =  y  —  x  -,  - . 

26.    Find  the  intercepts  of  the  tangent,  the  subtangent,  and  the  sub- 
normal of  the  parabola 

2/2  —  ^dx. 


CHAPTER  VI. 
THEORY  OF  EQUATIONS. 

86.  An  expression  of  the  form 

ax" -\- bx"-' +  cx''-^ -\-  .  ,  ,   -i-kx  +  l,  (1) 

where  n  is  a  finite  positive  integer  and  the  coefficients 
a,  b,  G,  .  .  .  k,  I  do  not  contain  x,  is  called  a  Rational  and 
Integral  Algebraic  Function  of  x  of  the  nth  degree;  and 

ax" -^  bx"-' +  Gx"-' -^  .   .   ,    ^kx-\-l=0  (2) 

is  called  the  General  Equation  of  the  nth  degree.  This  is  the 
kind  of  equation  we  shall  consider  in  this  section. 

If  we  divide  the  left  side  of  equation  (2)  by  a,  the  coefficient 
of  x",  we  shall  obtain  the  general  equation  of  the  nth  degree  in 
the  standard  form, 

X"  -\- p,X"-'  -^  P^""-'  +     .     .     .      H-pn-i^+i)n=0,  (3) 

where  p^,  p^,  .  .  .  p^-i,  p„  do  not  contain  x,  but  are  otherwise 
unrestricted.  As  will  be  seen  hereafter,  some  of  the  properties 
of  equations  can  be  stated  more  concisely  when  the  equation  is 
in  the  standard  form. 

In  this  section  the  symbol  f(x)  will  be  used  to  denote  a  rational 
integral  function  of  Xy  such  as  (1)  or  the  left  member  of  (3). 

Any  quantity  which  substituted  for  x  mf(x)  makes /(ic)  van- 
ish is  called  a  Root  of/(^);  or  a  Root  of  the  Equation 

f(ix)=0. 

If  we  put  y  =j{x)  and  plot  the  locus  of  this  equation,  we  shall 
obtain  a  curve  which  is  called  the  Graph  oif(x).  The  real  roots 
of  fix)  are,  therefore,  the  x-intercepts  of  its  graph. 

87.  A  rational  integral  function  of  x  is  continuous,  and  finite  for 
any  finite  value  of  x. 

Let    f(ix)=poX"+p,x"-'+P2X"-''-{-  .  .   .   +  p»-iX -{- pr,.     (1) 


88.]  THEORY    OF    EQUATIONS.  125 

Then  each  term  will  be  finite,  provided  x  is  finite ;  and  there- 
fore, as  the  number  of  terms  is  finite,  the  sum  of  them  all,  that 
is/(ic),  will  be  finite  for  any  finite  value  of  x. 

Now  suppose  X  receives  a  small  increment  A,  producing  in  /(a?) 
the  increment /(ic  -j-  K)  — f{x)  ;  then 

j{x^h)—f{x)=^P,[{x^hy—x'^-]+p,i{x^-hy-'-x-'^ 

+  .   .   .    J^p^_,l{x-\-h)—x-\.     (2) 

Each  of  the  terms  in  the  right  member  of  (2)  will  become  in- 
definitely small  when  h  is  indefinitely  small ;  hence  their  sum  will 
become  indefinitely  small.  Therefore  f{x  +  ^)  — fix)  can  be 
made  as  small  as  we  please  by  making  h  sufficiently  small.  This  shows 
that  as  X  changes  from  any  value  a  to  another  value  b,  f{x)  will 
change  gradually  and  without  interruption,  i.  e.  without  any  sud- 
den jump,  from /(a)  to/(6)  ;  so  that /(a?)  must  pass  at  least  once 
through  every  value  intermediate  to  /(a)  and  /(6).  That  is,  f{x) 
is  a  continuous  Junction. 

Hence  the  graph  of  f^x)  is  a  continuous  curve  with  finite  ordinates 
for  finite  values  of  X.  ' 

88.     To  calculate  the  numerical  value  off{a). 

Jjetfix)  =Pox'  ^PiX^-\-p.yX  -\-p3.  (1) 

Then  we  wish  to  calculate  the  numerical  value  of 

/(a)  =p^a^  -I-  p^a'  +  p^a  +  p,.  (2) 

This  result  is  most  easily  obtained  as  follows : 
Multiply  ^0  by  a  and  add  to  p^,  this  gives  p^a  +  p^ ; 
Multiply  this  by  a  and  add  to  p2,  this  gives  p^a"^  +  p^a  -\-  p^ ; 
Multiply  this  by  a  and  add  to  7)3,  this  givespo^^  +  i^i^^  H-i>2«  +^^3- 
The  process  may  be  arranged  in  the  following  way : 

i>o  Vx  V2  Ps 

Pj^ PoO^+Pia Poa^  +  Pi^^  +  P/^ 

Po  PiP'  +  Py  P<fl^'  +  jPitt  +  Pi  i>oa'  -h  i?i«'  +  i>2«  +  Pi- 

We  may  proceed  in  the  same  way,  whatever  the  degree  oif{x). 


126  THEORY    OF    EQUATIONS.  [89. 

Ex.    Find  the  numerical  value  of /(3)  if 

/(x)  =  2x*  —  7x3  +  13a,  _  jg^ 

2        —7  0  13        —16 

6       —3        —9  12 


_1        _3  4         _4 

.••    /(3)  =  -4. 

This  process  is  called  Synthetic  Substitution. 

89.  To  find  the  remainder  and  the  quotient  when  f{x)  is  divided  hy 
X  —  a,  where  a  is  any  constant. 

Divide /(a:)  hy  x  —  a  until  the  remainder  no  longer  contains  x. 
Let  Q  denote  the  quotient  and  R  the  remainder.     We  then 
have  the  identical  equation 

/(^)  =  §(^-a)+i?,  (1) 

which  must  be  satisfied  when  any  value  whatever  is  substituted 
for  X.     Let  x=^a^  then 

Ka)  =  Q(ia  —  a)^R  =  R',  (2) 

for  Q(a  —  a)=rO,  since  by  §  87  Q  remains  finite.  That  is,  the 
remainder  is  equal  to  the  result  obtained  by  substituting  a  for  x 
in  the  given  function. 

CoR.     If  a,  is  a  root  of  f(x),  thenf(x)  is  divisible  by  x  —  a. 

Conversely,  if  fix)  is  divisible  by  x  —  a,  then  a  is  a  root  of  f(x). 

For,  if  either /(a)  =0,  or  R  ^=0,  in  (2)  the  other  is  also  equal 
to  zero,  which  proves  the  proposition. 

Let  f(x)=pQX^-\-piX'^-{-p2X-\-p^,  for  example. 

By  actual  division  we  find 

Q  =PoX^  +  (i?oa  +  Vi)^  +  (Poa'+  i>ia  -[-P2), 

and  R  =poa^  +  p^a^  +  _P2«  +  Ps- 

By  comparing  these  expressions  with  the  results  found  in  g  88 
we  see  that  R  and  the  coefficients  in  Q  are  the  same  as  the  sums 
obtained  by  synthetic  substitution. 


90.]  THEORY  OF   EQUATIONS.  127 

Ex.    Find  Q  and  R  when  3ar'  —  2x*  —  16x3—  x  +  7  is  divided  by  [x  +  2). 

3        —2        —16        0       —1        +7 
—  6        +16       0  0        +2 


3—8  0       0       —1        +9 

Thus  Q  =  3x*— 8x3— 1,    ^nd    E  =  9. 

.-.    3x5— 2x*— 16x2 -x  +  7:e(x  +  2)(3x*  —  8x'—1)  +  9. 

This  process  can  be  applied  to  any  function  of  any  degree,  and 
is  a  particular  case  of  Synthetic  Division.  (See  Todhunter's 
Algebra,  Chap.  LVIII.) 

90.     An  equation  of  the  nth  degree  has  n  roots,  real  or  imaginary. 
Let  the  equation  be 

fix)=ar-^p^x"-'-\-p,x-'-i-  .  .  .   +p^  =  0.  (I) 

Let  tti  be  one  root*  of  the  equation  /(x)  =  0,  then  f(x)  is  divis- 
ible by  (aj— a,).     (§89.) 

...    f(x)  =  (x-a,)f,ix),  (2) 

where /i(^)  is  an  integral  function  of  x  of  degree  (n  —  1). 
In  like  manner  if  a2  is  a  root  of  Ji{x),  then 

S,{x)^{x  —  a,)j,{x),  (3) 

\Ybere/2(a?)  is  an  integral  function  of  x  of  degree  (ti  —  2). 

Proceeding  in  this  way  we  shall  find  n  factors  of  the  form 
{x  —  a^),  and  we  have  finally, 

f{x)  =  {x  —  a,){x  —  a2){x  —  a^)  .   .   .   {x  —  a„)=0.     (4) 

It  is  now  clear  that  a,,  ag,  ag  .  .  .  a„  are  roots  of  the  equation 
f(^x)  =0;  and  as  no  other  value  of  x  wiU  make /(a?)  vanish,  the 
equation  can  have  no  other  roots. 

The  factors  of  f{x)  need  not  all  be  different  from  one  another ; 
thus  we  may  have 

J{x)  =  {x  —  a,y(,x  —  a.;)'^{^x—a,y  .   .   .   ,  (5) 

where  'p-\-q-\-r-\-...=n. 

*We  here  assume  the  fundamental  theorem  that  every  equation  has  one  root,  real  or 
imaginary'.  Proofs  of  this  theorem  have  been  given  by  Argand,  Cauchy,  Clifford,  and 
others,  but  they  are  too  difficult  to  be  included  in  this  book.  The  student,  however,  is 
already  familiar  with  the  fact  that  every  equation  of  the  first  degree  has  one  root;  that 
every  equation  of  the  second  degree  has  two  roots,  real  or  imaginary;  and  it  will  be  shown 
in  §  94  that  every  equation  of  an  odd  degree  has  one  real  root. 


128  THEORY   OF   EQUATIONS.  [91. 

In  this  case /(a?)  lias^  roots  each  a^,  q  roots  each  ag?  etc.,  the 
whole  number  of  roots  being 

p-\-q-i-r-\-   .    .    .    =n. 

Therefore  the  graph  of  f(x)  will  cut  the  a^-axis  in  n  points, 
which  may  be  real,  coincident,  or  imaginary ;  and  the  real  roots 
are  its  a?-intercepts. 

Hence  the  real  roots  of  a  function  may  be  found  exactly  or  ap- 
proximately by  constructing  its  graph. 

EXAMPLES. 

1 .  Divide  2x^  —  6x*  —  Sx^  +  lOx  +  18  by  x  —  3. 
Find  the  other  roots  of  the  following  equations : 

2.  Two  roots  of  x*  —  12x3  +  49^2  _  78x  +  40  =  0  are  1  and  5. 

3.  One  root  of  x^—  IQx^  +  20x  +  112  ==  0  is  —  2. 

4.  Two  roots  of  x*  +  8a^  —  22^^  —  16x  +  40  =  0  are  2  and  —  10. 

5.  Two  roots  of  x*  —  12x3  +  48x2  —  68x  +  15  =  0  are  5  and  3. 

6.  Three  roots  of  6x»  +  Ux*  —  21x3  +  Vx^  +  15x  — 18  =  0  are  d=  1  and  —  3. 
Find  graphically  the  exact  or  approximate  roots  of 

7.  x3  — 2x2  — llx  + 12  =  0. 

8.  X*  — 8x3+14x2  +  8x  — 15==0. 

9.  X*  — 2x3— 13x2  — 14x  + 24  =  0. 

10.  x3  — 8x2  — 28x  + 80=0. 

11.  6x3— 13x2  — 21x  + 18=0. 

12.  8x3— 18x2  — 71x  + 60  =  0. 

13.  X*— 6x3  — 5x2  +  56x  — 30  =  0. 

91.     Relations  between  the  roots  and  the  coefficients  of  an  equation. 
If  there  are  two  roots,  a^  and  aa,  we  have  (§90) 

X^  +  PiX  +  ;)2  =  (^ «i)  (OC ttg) 

=  x'^ — («!  +  aa)^?  +  aittg.  (1) 

.-.     a,^a2=—p„         a,a^=p^. 
If  there  are  three  roots  a^,  a^,  and  ag,  we  have 

x^  +  PxX^  -\- p^x  ^  p^=  {x  —  tti)  {x  —  a^){x  —  ag) 

=  x^ —  (di  +  a2  +  a^)x^  +  (aitta  +  «2«^3  +  (^zO'\)x  —  a^a^a^     (2) 

,  •  .        ai  +  ttg  +  ^3  = P\1         «ia2  +  a2<*3  +  "sO^l  =  ?2J  (^xf^lff'Z  =  —  i>3- 


^1.]  THEORY    OF    EQUATIONS.  129 

In  like  manner  if  the  equation  is  of  the  nth  degree  and  there- 
fore has  n  roots  ai,  aj  .   .   .  a^  .   .   .  a,.,  then 

=  {x  —  ai)(ix  —  a,)  .   .   .    (x  —  a,.)   .   .   .   (x  —  a,)     (3) 

=  x'*—s,x^-'-\-s.:^''-'—  .  .  .  +(~iys^-'- 

±  .   .   .   +(-ir>S.,     (4) 

where  S^  is  the  sum  of  all  the  products  of  a^,  ao,  .  .  .  a^  .  .  .  a„ 
taken  r  together. 

Equating  the  coefficients  of  the  same  powers  of  x  on  the  two 
sides  of  the  identity  (4)  gives 

Si  =  —P„   s,=p,,   s,=  (—iypr, 

iS„=(— l)"p„=(— l)"a,a,,  ...   a,  ...   a„. 

The  absolute  term,  p,„  is  divisible  by  each  of  the  roots. 
If  J),,  =0,  one  root  is  zero;  if  p„  ^ p„_i=:  0,  two  roots  are  zero;  if 
j9^,=  j9^_jrr=:  .  .  .  p^_^=zO,  r  -{- 1  roots  are  zero. 

EXAMPLES. 
Find  the  other  roots  of  the  following  equations : 

1.  Two  roots  ofx^-f-  x^  —  ix  —  4  =  0  are  2  and  —  1. 

2.  Two  roots  ot  a^  —  ix'  —  3x  -\-i2  =  0  are  4  and  /3. 

3.  Two  roots  of  x'  —  13x  +  12  =  0  are  1  and  3. 

4.  Three  roots  of  a^  —  lOa^  +  35x2  _  sOx  +  24  =  0  are  1,  2,  and  3. 

5.  Onerootof  x3_43.2_a._ois  — (24-y'5). 

6.  One  root  of  af  —  6x*  +  12x3  =  0  is  3  —  V^S. 

7.  Two  roots  of  6x*  —  Tx^  —  14x2  +  15x  =  0  are  1  and  |. 

8.  Two  roots  of  4x5  —  5x*  +  2x3  +  Gx^  =  0  are  1  zb  V^^. 
Form  the  equations  whose  roots  are 

9.  1,3,-5.  10.    —2,3,-4,6. 
11.    h-hh                                       12.    ±1,±4. 

13.  0,1,-4,5.  14.  ±i/2,  4ri/3. 

15.  0,  -  2,  di  /=^.  16.  3,  5  ±  v/5. 

17.  4 zh i/3,  —  1  dr  v/6.  18.  1,-2,3,-4,5. 

19.  0,  2  ±  i/=3,  — 3  ±  ]/6.  20.  0,  0,  ^,  —  §,  1  =b  i/2. 

21.    l±l/^^,— 2=bi/^^.  22.    — 3,  2=bi/^,— 3d=l/^^. 

10 


130  THEORY    OF    EQUATIONS.  [94  ► 

92.  The  first  term  of  jXx)  can  be  made  to  exceed  the  sum  of  all  the 
other  terms  by  giving  to  x  a  value  sufficiently  great. 

Let         f{x)=p^''-\-p,x"-'-\-p<^''-^  +  .   .  .   -h;)„, 

and  let  k  be  the  greatest  of  the  coefficients ;  then 

l^ ^ 2^ 

^iaJ~-^-hi>2^"-'+  .  .  •   -^  Pn      Kx"-' -i- x^-' -\-  ...  4-1) 

P,XXX  —  1^        p^%X  —  l)  ^  ?o  ,     _  -,  X 

^     k{x"—l)   ^         kx''    .    ^  k^^       ^' 
Now  ^{x  —  1 )  can  be  made  as  great  as  we  please  by  sufficiently 
increasing  x,  which  gives  the  proposition. 

93.  An  even  number,  or  an  odd  number,  of  real  roots  off^x)  =0 
lie  between  a  and  b  according  asf{a)  and  f(b)  have  the  same  sign,  or 
opposite  signs. 

The  two  points  A[a,  /(a)]  and  -B[6,/(6)]  are  on  the  same  side, 
or  on  opposite  sides,  of  the  a::- axis  according  as /(a)  and/(6)  have 
the  same  sign,  or  opposite  signs. 

Therefore,  since  the  graph  of /(a?)  is  a  continuous  curve  (§87), 
in  passing  from  A  to  B  along  the  graph  the  a:;-axis  will  be  crossed 
an  even  number,  or  an  odd  number,  of  times  according  as /(a)  and 
f(6)  have  the  same  sign,  or  opposite  signs.  This  proves  the  propo- 
sition. 

E.  g.,  if  f(x)  =  x'-Sx-\-i,  then /(I)  =  —  1  and/(2)  =  3. 

.  •.    At  least  one  real  root  of  x^  —  3x -f- 1  =  0  lies  between  1  and  2. 

94.  An  equation  of  an  odd  degree  has  at  least  one  real  root. 
Let  the  given  equation  be 

f(x)=x"^-'+p,x'^-^p,x'"-'+  .  .   .   +i)2«^i=0. 

Let  a  be  a  value  of  x  sufficiently  large  to  make  the  first  term  of 
/(a)  greater  than  the  sum  of  all  the  other  terms  (§  92).  Then 
the  sign  of /(a)  will  be  the  same  as  the  sign  of  a^"  +  \  i.  e.  the  same 
as  the  sign  of  a. 

Hence  if  a  be  sufficiently  great,  /(a)  is  positive,  /(O)  =  |>2«  + 1?  ^-nd 
/( —  a)  is  negative. 

Therefore  in  all  cases  there  is  one  real  root,  which  is  positive  or 
negative  according  as|}2„+i  is  negative  ov  positive  (§  93). 


96.] 


THEORY    OF    EQUATIONS. 


131 


Hence  the  graph  of  a  function  of  an  odd  degree  in  the  standard  form 
extends  to  infinity  in  the  first  and  third  quadrants, 

95.  An  equation  of  an  even  degree  in  the  standard  form  with  the 
last  term  negative  has  at  least  two  real  roots  with  opposite  signs. 

Let  the  given  equation  be 

fiia0^x'"+p^x'^'-'^p,x'"-'+  .  .   .   +jp2n=0. 

If  a  is  taken  sufficiently  great,  /(a)  will  have  the  same  sign  as 
a'"  (§  92),  which  is  positive  for  both  positive  and  negative  values 
of  a;  that  is, /(a)  and/( — a)  will  both  be  positive,  while/(0)  =p2ny 
which  by  hypothesis  is  negative. 

Therefore  there  is  at  least  one  real  root  between  0  and  a,  and 
another  between  0  and  —  a  (■§  93). 

The  graph  of  a  function  of  an  even  degree  in  the  standard  form  ex- 
tends to  infinity  in  the  first  and  second  quadrants. 

96.  To  find  approximately  the  real  roots  of  f(^x)  =  0. 

Construct  the  graph  of  /(ic)  and  thus  determine  the  pairs  of 
consecutive  integers  between  which  the  roots  lie. 


Suppose  f{a)  =  CA,  a  positive  number;  and  f(a  4-  1)=DB,  a 
negative  number. 

Then  there  is  at  least  one  real  root  (§  93)  between  a  and  a  + 1. 

Draw  the  chord  AB  cutting  the  ic-axis  in  E;  draw  ^i^  parallel 
to  the  X'Sbxis  meeting  A  C  produced  in  F, 


132  THEORY    OF    EQUATIONS.  [97. 

Then,  if  there  is  only  one  root  between  a  and  a  -|-  1,  it  is  approxi- 
mately equal  to  OE;  if  the  graph  were  a  straight  line,  it  would  be 
exactly  equal  to  OE. 

Since  the  triangles  A  CE  and  AFB  are  similar, 

f,j,_FB-CA  CA  fja) 

FA  CA+BD     /(«)_/(« +!)•  ^'^ 

If  we  use  iiumeHcal  values  of  /(a)  and /(a  -|-  1),  we  shall  then 
have  for  all  cases 

^^  =  "+/(a)//(i+l)-  ^^> 

Ex.    Find  the  roots  of  x^  —  29x  +  42  =  0. 

Here /(4)  =  —  10  and  /(5)  =  22.  Hence  there  is  one  root  between  4  and  5. 
Substituting  in  (2)  gives 

<'^  =  ''+1o4r-2  =  ''-*- 

Then/(4.4)  =  —  .456  and/(4.5)  =  3.081. 

Hence  the  root  lies  between  4.4  and  4.5. 

When  the  root  is  greater  than  OE,  as  in  the  diagram  and  also  in  this  ex- 
ample, it  is  better  to  try  the  figure  next  greater  than  that  given  by  the 
quotient. 

The  next  figure  of  the  root  may  now  be  approximated  in  the  same  way. 

Thus  _^_/(4.4)        ^A56^oi 

^^"^^  /(4.4)+/(4.5)      3.537      •^'• 

,'.    The  approximate  root  is  4.41.    The  exact  root  is  (3  +  \'"2,). 

EXAMPLES. 

Calculate  to  two  places  of  decimals  the  real  roots  of  the  following  equa- 
tions : 

1.  x3-3x  — 1=0.  2.  x'^-7x 4-7  =  0. 

3.  x3  +  2x'^  — 3x  — 9  =  0.  4.  x5  +  2x2  — 4x  — 43  =  0. 

5.  x3  — 15x  +  21=0.  6.  X*— 122;  +  7  =  0. 

7.  X*  — 5x^  +  2x2— 13x4-55  =  0.  8.  x^  — 3x^  —  2x4-5=0. 

9.  x5  — 81x4-40  =  0.  10.  X*  — 55x2  — 30x+ 400  =  0. 

97.  In  any  equation  with  real  coefficients  imaginary  roots  occur  in 
pairs, 

I.  Let/(a^)  =  0  be  an  equation  with  real  coefficients  having  r 
real  roots  and  the  other  roots  imaginary.     Then 

f(x)^(x  —  a,)(ix—a,)  ,  .   .   (x  —  a,)<p(x)=0,       (§90)       (1) 


97.]  THEORY    OF   EQUATIONS.  13S 

where  <p{x')  is  a  function  with  real  coefficients  whose  roots  are  all 
the  imaginary  roots  of  /(d?),  and  no  others.     Hence  <f{x)  must 
be  of  even  degree,  and  therefore  has  an  even  number  of  roots. 
Otherwise  it  would  have  at  least  one  real  root  (§  94). 
Therefore  (1)  has  an  even  numher  of  imaginary  roots. 

II.  If  a  -\-  hV  —  i  is  a  root  of  an  equation  with  real  coefficients^ 
then  a —  bv  —  1  is  also  a  root. 

Let  the  equation  be 

£c''+p,^"-^+i>2^"-^+  .   .  .   -h;)„-=0.  (2) 

Substituting  a  -{-  bV  —  1  for  j::  in  (2),  we  have 

(a  +  6l/^=^)"  +p,(a  -h  61/^^1)—  -\- 2),(a  -j-  bV^^)x''-' 

-f-  .   .   .    ^p,,=  0.     (3) 

Expanding  by  the  binomial  theorem,  and  collecting  together 
the  real  and  imaginary  terms,  we  shall  have  a  result  in  the  form 

F  +  QV~=^1  =  0.  (4) 

In  order  that  this  equation  may  hold  we  must  have 

P=Q  =  0.  (5) 

Since  Pand  Q  are  realj  they  contain  only  even  powers  of  l/  —  1, 
and  hence  will  not  be  changed  by  changing  the  sign  of  l/ —  1. 
Therefore,  when  a  —  6l/ —  1  is  substituted  for  a:  in  (2),  the  re- 
sult will  be  P  —  QV^^. 

But  from  (5)  P  —  QV^^  =  0. 


.-.     a  —  bV — 1  is  also  a  root  of  (2). 

Corresponding  to  the  roots  a  ±  bV —  1  of  J{x)  =  0,  f(x)  will 
have  the  real  quadratic  factor  [(x  —  a)-  -{-  b-]. 

The  two  quantities  a  ±  6l/ —  1  are  called  conjugate  imaginary 
expressions. 

Show  that  the  locus  of  the  equation  y  =  x^  -{-  k  cuts  the  ^r-axis 
in  trvN^o  points  which  are  real  and  distinct,  real  and  coincident,  or 
imaginary  according  as  k  is  negative,  zero,  or  positive.  Hence 
illustrate  graphically  the  preceding  theorem  by  showing  that,  as 
the  absolute  term  oif(x)  is  changed,  real  intersections  of  its  graph 


134  THEORY    OF    EQUATIONS.  [98. 

with  the  a?-axis  disappear  or  reappear  in  pairs ;  and  that  the  pas- 
sage from  a  pair  of  real  distinct  roots  to  a  pair  of  imaginary  roots 
is  through  a  pair  of  real  coincident  roots. 

EXAMPLES. 

1.  Show  that  if  either  a  ±  \/h  is  a  root  of  an  equation  with  rational  co- 
efficients, the  other  is  also  a  root. 

2.  Solve  the  equation  x*  —  2x^  —  22x'  -\-  62x  —  15  =  0,  having  given  that 
one  root  is  2  +  y  3. 

3.  Solve  the  equation  2x^ —  15x^  +  ^^^  —  42  =  0,  having  given  that  one 
root  is  3  +  V^^h. 

4.  If  |/a  +  ^/6  is  a  root  of  an  equation  with  rational  coefficients,  y  a  and 
l/6  not  being  similar  surds,  show  that  ±  y  a  ±  yh  will  all  four  be  roots. 

5.  Form  the  biquadratic  equation  with  rational  coefficients  one  root  of 
which  is  i/2  +  v/3. 

6.  Show  that  Ex.  4  holds  when  either  or  both  a  and  h  are  negative. 

7.  Find  the  biquadratic  equation  with  rational  coefficients  one  root  of 
which  is  /2  +  V^^'6. 

8.  Solve  the  equation  2x^  —  3a;^  +  5x*  +  6x^  —  27x  +  81=0,  having  given 
that  one  root  is  \/2-\-V —  1. 

Transformation  of  Equations. 

98.  I.  To  find  an  equation  whose  roots  are  those  of  a  given  equation 
'with  opposite  signs. 

If  the  given  equation  is/(a:;)  =  0,  the  required  equation  will  be 
/( — x)  =  0.  For,  when  x  =  a,  f(x)=f(a) ,  and  when  x=  —  a, 
/( —  x)=f(a)  ;  hence,  if  <x  is  a  root  of /(^)  =  0,  then  —  a  will  be 
a  root  of  /( —  x)  =  0. 

The  graph  of /( —  x)  is  the  reflection  of  the  graph  oi /(x)  in  a 
mirror  through  the  7/-axis  perpendicular  to  the  plane ;  i.  e.  the  two 
graphs  are  symmetrical  with  respect  to  the  7/-axis,  which  proves 
the  transformation  for  real  roots. 

I^/(^)  =/( —  ^)  [§  28,  (2)],  the  two  graphs  will  coincide,  and 
the  roots  of  f{x)  will  occur  in  symmetric  pairs  of  the  form  ±  a. 

If  the  equation  is  complete,  this  transformation  is  effected  by 
simply  changing  the  sign  of  every  other  term  beginning  with  the 
second. 


98.]  THEORY   OF    EQUATIONS.  135 

II.  To  find  an  equation  ivhose  roots  are  those  of  a  given  equation,  each 
diminished  by  the  same  given  quantity. 

If  we  put  X  =  x'  -\-  h,  the  origin  will  be  moved  to  the  right  a 
distance  equal  to  h  [§  66,  (10)]. 

Hence  the  a^-intercepts  of  the  graph  of  /(x),  i.  e.  the  real  roots 
of /(a*),  will  each  be  diminished  by  h. 

Therefore,  iif(x)=0  is  the  given  equation,  the  required  equa- 
tion will  be  f(x  +  h)=0.  For,  when  x  =  a,  f{x)  ^f{a)^  and 
when  J-  =  a  —  h^  f(x  +  h)  =f(a) ;  hence,  if  a  is  a  root  of  f(x)  =  0, 
then  a  —  his  also  a  root  of  f(^x  -\-h)=0,  whether  a  is  real  or 
imaginary. 

The  coefficients  of  the  new  equation  can  be  found  by  synthetic 
substitution  as  follows : 

Ex.    Find  the  equation  whose  roots  are  those  of 
x*  —  3x'  —  15x'-^^9x  —  i2  =  0 
each  diminished  by  2. 

Operation 


1 

—  3 
2 

-15 
-2 

+  49 
-34 

-12 
+  30 

i 

—  1 
2 

-17 
2 

H-15 
—  30 

+  18 

1 

+  1 
2 

—  15 

+  6 

-15 

1 

+  3    , 

■2    1 

-9 

5 

X*  +  5x^  —  9x-  —  15x  +  18  =  0  is  the  required  equation. 


If  we  put  x  =  x' — -,  where  «,  is  the  coefficient  of  oc"~\  each 
n 

root  will  be  diminished  by  I  —  —  j,  and  therefore  the  sum  of  the 

roots  will  be  diminished  by  n  I  —  —  )==  — Pi- 

Hence  the  sum  of  the  roots  of  the  new  equation  will  be  zero  (§  91); 
i.  e.  the  coefficient  of  the  second  term  will  be  zero. 

Ex.    Transform  the  equation  x'  +  6x2  +  4a;  +  5  =  0  into  another  in  which 
the  coefficient  of  x'^  is  zero. 


136  THEORY   OF   EQUATIONS.  [98. 

Let  X  =  x' —  2,  since  pi=  6  and  n  =  3;  then  we  obtain 


1 

t\ 

n 

+  5 
+  8 

1 

t\    z\ 

+  13 

1 

n 

-8 

1 

0 

, 

.    a^-8x+13  = 

=  0  is  the  required  equation. 

III.     To  find  an  equation  whose  roots  are  the  reciprocals  of  the  roots 
of  a  given  equation. 

Let  the  given  equation  be 

P^''-\-PlX''-'-\-p2X''-'-\-     .      .      .      ■^Pn-iX-\-p,=  0.  (I)' 

Substituting  —  for  a?  in  (1)  gives 


H9"+^<9''"+49"'+-  •  +^^-<9+^"='' 


(2) 


which  is  the  required  equation,  for  (2)  is  satisfied  by  the  recip- 
rocal of  any  quantity  which  satisfies  (1). 
Multiplying  (2 )  by  z"  gives 

p.s"+i>«-,2"-'  +  i5«-22"-'+  .   .   .   -\-p,^+Po  =  0.  (3) 

Therefore  the  required  equation  is  obtained  by  merely  reversing 
the  order  of  the  coefficients  of  the  given  equation. 

li  p^  =  0,  one  root  of  (1)  is  zero,  and  hence  the  corresponding^ 
root  of  (2  )  is  infinite.  Therefore,  as  the  coefficient  of  the  highest  power 
of  X  inf(x)  approaches  zero,  one  root  of  f(x)  approaches  infinity. 

If  the  coefficients  of  (1)  are  the  same  (or  differ  only  in  sign) 
when  read  in  order  backwards  as  when  read  in  order  forwards, 
the  roots  of  (1)  and  (3)  are  the  same.     That  is,  the  roots  of  (1) 

will  then  occur  in  pairs  of  the  form  a  and  -. 

a 

An  equation  in  which  the  reciprocal  of  any  root  is  also  a  root 
is  called  a  Reciprocal  Equation. 

E.  g.j  Qx^ —  19x^  +  19x  —  6  =  0  is  a  reciprocal  equation  in  which  the  co- 
efficients differ  in  sign  when  read  in  order  backwards  and  forwards ;  two 
roots  are  |  and  f . 


98.]  THEORY   OF    EQUATIONS.  137 

EXAMPLES. 

Find  the  equations  whose  roots  are  those  of  the  following  equations  with 
opposite  signs : 

I.  x^  — 4x  — 5  =  0.  ^  2.    x^  +  6x2  — 7x  — 60=T). 
3.    x3_8a.2_28x  +  80=0.  4.    x*— 12x2  + 12x  — 3  =  0. 
Find  the  equation  whose  roots  are  those  of 

5.  x»  —  16x2  +  20x  +  112  =  0,  each  diminished  by  4. 

6.  X*  —  12x3  +  49x2  —  78x  +  40  =  0,  each  diminished  by  2 . 

7.  X*  —  3x3  —  6x2  4-  14x  +  12  =  q,  each  diminished  by  -  2. 

Transform  the  following  equations  so  as  to  make  the  second  terms  dis- 
appear: 

8.    x2  — 4x  — 21=0.  9.    x3  — ex'H-Sx  — 2  =  0. 

10.    x*  +  4x3  — 29x2  — 156x+ 180  =  0. 

II.  Find  the  equation  whose  roots  are  those  of  x^  +  6x2  —  i^^  -f- 12  =  0 
each  diminished  by  c,  and  find  what  c  must  be  in  order  that,  in  the  trans- 
formed equation,  (1)  the  sum  of  the  roots,  and  (2)  the  sum  of  the  products 
of  the  roots  two  together,  may  be  zero. 

12.  Transform  the  equation  xr^-\-Sx^  —  9x  —  27  =  0  into  another  in  which 
che  coefficient  of  x  shall  be  zero. 

Find  the  equation  whose  roots  shall  be  the  reciprocals  of  the  roots  of 

13.  x2  — 8x  — 9  =  0.  14.    2x3  +  3x2  — 13x  — 12  =  0. 

15.  6x*  —  5x3  _  30^2  +  20x  +  24  =  0. 

16.  Show  that  a  reciprocal  equation  of  an  odd  degree  whose  correspond- 
ing coefficients  have  the  same  sign  has  one  root  equal  to  —  1. 

17.  Show  that  a  reciprocal  equation  of  an  odd  degree  in  which  corre- 
sponding coefficients  have  opposite  signs  has  one  root  equal  to  +  1. 

18.  Show  that  a  reciprocal  equation  of  an  even  degree  in  which  corre- 
sponding coefficients  have  opposite  signs  has  the  two  roots  ±  1. 

Solve  the  following  equations: 

19.  2x^  — 7x2  +  7x-2  =  0.  20.    6x3  — 7x2 -7x  + 6  =  0. 
21.    3x3  +  5x2  +  5x  +  3  =  0.  22.    5x3  — 7x2  +  7x  — 5  =  0. 

23.    2x^  +  5x3-5x2  —  2  =  0.  24.    12x<  —  25x3  +  25x  — 12  =  0. 

25.  6x*  — 7x3  +  7x-6  =  0. 

26.  Solve  the  equation  2x*  —  3x3  —  iq^2  _  33;  _j_  2  =  0,  having  given  that 
one  root  is  —  2. 

27.  Solve  the  equation  Ux^  —  3x*  —  34x3  —  34x2  —  3x  + 14  =  0,  having 
given  that  one  root  is  2. 

28.  Solve  the  equation  10x«  —  21x^  +  21x  —  10  =  0,  having  given  that 
one  root  is  2. 


138 


THEORY    OF    EQUATIONS. 


[100. 


99.  Successive  Derivatives.  If  f(x)  denote  any  function  of  x, 
its  derivative /(a?),  (§  79),  will  in  general  be  a  function  of  x  that 
can  also  be  differentiated.  The  result  of  differentiating /'(^)  is 
called  the  Second  Derivative  of  f(x).  U  this,  again,  can  be 
differentiated,  the  result  is  called  the  Third  Derivative,  and 
so  on. 

The  successive  derivatives  of  f(x)  will  be  denoted  by 
fU),     r(x),     /'"(;/■)   .   .   .  .p)(.r). 

Let        f(x)  E  A,  +  J  ,,r  +  A,x'  +  ^3^^ 
Then    /'(.r)  =  A,  +  2Aa'  +  SA,x'  +  . 
r(x)=2A,-^2'^A,x-\-  .   .   . 


.   .   .    -\-A„x\ 
.   -^  7iA,,jc^'-\    (§83) 
n(n  —  1  )A„x"~'^, 


/'"(^)=l-2-3^+   .    .   .    -f,i(,i  — l)(n  — 2)X^."-^ 

f^\x)  =7i(n  —  l)(n  —  2)   .  .    .    -^S'2'  lA„=A,rn\. 
E.  g.,  if  f{x)  =  x*  —  Sx'  —  5x^  +  2x  —  1, 
then       /(x)  =  4x^  —  9x- —  lOx  +  2,          •  fix)=24:X  —  18, 

///(x)  =  12^2  —  18x  —  10,  f'ix)  =  24  =  4 ! . 

Hence  the  rth  derivative  of  a  rational  integral  function  of  the 
tith  degree  is  itself  a  rational  integral  function  of  degree  (n  —  r), 
(where  r  is  not  greater  than  n)  ;  and  the  nth.  derivative  is  a  con- 
stant. Therefore  the  preceding  theorems  pertaining  to  a  rational 
integral  function /(^)  will  also  hold  for  its  derivatives. 


100. 

The  Derivative  Curve, 

and  Elbows. 

Y 

T 

L 

\ 

/ 

\     B 

,' 

15.^ 

y/ - 

V 

a'/' 

q/    \i 

\o        cV 

/' 

L^. 

5 

<U 

A 

r     \ 

'''     0 

/h 

D' 

Let  the  curves  iJ/and  i'J/'be  the  loci,  respectively,  of  the 
equations 


and 


/(^) 
/(^). 


(1) 
(2) 


100.]  THEORY    OF   EQUATIONS.  139 

We  will  call  L'M',  the  locus  of  (2),  the  Derivative  Curve, 
(or  D.  C),  and  LJ/the  Integral  Curve.     (See  §  102.) 

Draw  any  line  parallel  to  the  7/-axis  meeting  the  a;-axis  in  §, 
and  the  curves  in  Pand  P'. 

We  will  call  Pand  P'  corrssponding  points. 

Then,  if  0§  =  a,  we  have  b}'  §  79 

QP'  =  /'(a)  =  slope  of  LM  at  P. 

Hence  the  D.  C.  is  a  curve  such  that  its  ordinate  at  any  point 
is  the  slope  of  the  integral  curve  at  the  corresponding  point. 

Let  A,  B,  C,  D  be  the  points  on  LJ/ where  the  slope,  i.  e.  /'(a;), 
is  zero;  then  the  ordinates  of  the  corresponding  points  ^',  P',  C, 
ly  on  L'W  are  zero.  Hence  J.',  P',  C",  D'  are  the  intersections 
of  L'M'  with  the  a?-axis.  Between  A  and  B  the  slope  of  LM  is 
positive,  between  B  and  C  negative,  etc.  Therefore,  between  A' 
and  P'  the  curve  L'M'  is  above  the  a?-axis  between  B'  and  C 
below,  etc. 

It  will  be  convenient  to  call  such  points  as  A  P,  C,  P,  Elbows 
of  the  curve.  Then  the  abscissas  of  the  elbows  of  the  graph  of 
/(j")  are  the  roots  of  /'(.r),  and  may  therefore  be  found  by  plot- 
ting the  D.  C.  or  by  solving  the  equation /'(j^)  =  0. 

Since /'(a?)  is  of  degree  {n —  1),  (§99,)  the  graph  oif{x)  can 
not  have  more  than  (?i  —  1)  elbows. 

If /(d?)  is  of  an  odd  degree,  its  graph  will  have  an  even  number 
of  elbows,  and  therefore  f(x)  will  have  at  least  one  real  root, 
(tj.  §94.) 

If  the  roots  oif'(x)  are  imaginary,  the  graph  oif{x)  will  have 
no  elbows. 

If  two  roots  oif{x)  are  equal,  its  graph  will  touch  the  a?-axis, 
as  at  D',  and  the  two  corresponding  elbows  of  the  integral  curve 
will  coincide  as  shown  at  D.  Hence  the  slope  of  LM  has  the  same 
sign  on  both  sides  of  P.  The  integral  curve  therefore  changes 
the  direction  of  its  curvature  at  P,  and  crosses  its  own  tangent, 
which  it  cuts  in  three  coincident  points.  Such  a  point  is  called  a 
Point  of  Inflection. 

Ex.    Find  the  coordinates  of  the  elbows  of  the  following  loci: 
1.    y  =  x'—i2x.  2.    y  =  2x^—i5x--\-2ix-\-5. 

3.    2/  =  a^  — 6x2  +  32.  4.    2/ ==3:r^— 20r"'+ 18a"^-h  108x. 

5.    2/ =  30^  —  20x^+10.  6.    y  =  Sx*~S3T^  —  66x'-{-iUx. 


140  THEORY   OF   EQUATIONS.  [101- 

Equal  Roots 
101.     Rolle's  Theorem.     At  least  one  real  root  of  the  equation 

/'(.r)=0  (1) 

lies  between  any  two  consecutive  real  roots  of 

/Ct)=0..  (2> 

For  there  is  at  least  one  elbow  of  the  integral  curve,  LM  (§  lOO), 
between  any  two  consecutive  intersections  of  it  with  the  x-Sbxis. 

Conversely,  LM  ca>n  not  meet  the  a:-axis  more  than  once  be- 
tween any  two  of  its  consecutive  elbows. 

Therefore,  at  most  one  real  root  of  (2)  lies  between  any  two 
consecutive  real  roots  of  ( 1 ) . 

That  is,  the  real  roots  of  (1)  separate  those  of  (2). 

If  by  a  continuous  modification  of  the  form  oif(x) — for  example, 
by  the  addition  or  subtraction  of  a  constant  (§  97) — two  roots  are 
made  equal,  the  root  oif(x)  lying  between  them  must  approach 
the  same  value.     Hence  a  double  root  of  (2)  is  also  a  root  of  (1). 

In  general,  if /(^r)  has  an  r-fold  root,  such  a  root  being  regarded 
as  due  to  the  coalescence  of  r  distinct  roots,  then  will  f(x)  have 
an  (r  —  l)-fold  root  due  to  the  coalescence  of  the  (r  —  1)  inter:* 
vening  roots.  That  is,  if  f(x)  has  r  roots  each  equal  to  a,  f{x) 
will  have  (r  —  1)  roots  each  equal  to  a. 

Then,  by  the  application  of  Rolle's  theorem  tof(x)  and/"(a?)y 
f"{x)  and /'"(a?),  and  so  on, 

if  f(ix)  =  (x  —  ay<p{x), 

we  have  f(^)  =  (^  —  o^)'"~Vi(a^)> 


rCr)  =  Cr-ay-Y,Cx) 


f'-^\jc)  =  (x~a)<p^._,(x). 


(3) 


Conversely,  if  r  roots  of  /'(.r)  coalesce  and  become  equal  to  a,, 
the  corresponding  r  elbows  of  the  integral  curve  LM  will  coalesce ; 
then,  if  a  is  a  root  oif(x),  this  r-fold  elbow  will  rest  on  the  a:'-axis 
and  give  an  (r  +  l)-fold  root  oif(x). 

Hence  by  induction,  if 

/('-')(«)  =/<•■- ^'(«)  =/^'-'  («)  =   ■   •    .  /'(a)  =/(«)  =  0, 


101.]  THEORY    OF    EQUATIONS.  141 

and  a  is  a  single  root  of  f^^~^^{x),  then  a  is  a  double  root  of 
/''-2)(ic),  a  triple  root  of /('•-^JCic),  .  .  .  an  (r  —  1) -fold  root  of 
f(x)j  and  an  r-fold  root  oi  f(x). 

Ihis  suggests  an  easy  method  of  finding  real  multiple  roots  of 
an  equation,  when  the  roots  are  all  equal  except  one  or  two. 

E.  g.,  if  fix)  =x'  —  5x*  +  dOx'^  —  80x  +  48  =  0, 

we  have  f{x)=  5x*  —  2(k^  +  SOc  —  80, 

f'{x)  =  20r*  —  60x'^  H-  80, 
fix)  =  60x'  —  12ac. 

The  roots  of  eOx'  —  120^  =  0  are  0  and  2. 

Since  /'^'(2)  =f'i2)  =/(2)  =/(2)  =  0,  2  is  a  fourfold  root  of  fix)  =  0. 
Hence  all  its  roots  are  2,  2,  2,  2,  — 3. 

Equations  (3)  are  true  whether  a  is  real  or  imaginary.  For 
suppose /(a?)  has  an  r-fold  root  equal  to  a,  then,  whether  a  is  real 
or  imaginary,  we  have  (§89  and  §  90) 

Kjc)^ix-ay<pix).  (4) 

Then 

Expanding  [(a^  —  a)4-  fi]''  by  the  binomial  theorem  gives 

lim    ([ix  —  ay-\-rix  —  ay-'^h]0(^x-\-h) 

/I  ) 

+  [r(x  —  a)'-i  +  Mr  —  i){x  —  a,^-'h  +  .  .  +  /i'-^]©(x  +  h)  I      (7) 

=  (x  — a)'-?»^(a:)  +  r(x  — a)'-i^(x)  (8) 

=  (X  — ar-'Clx  — a)^^(x)  +  r?>(x)]  =(a;  — ar-ViCx),  (9) 

which  is  of  the  same  form  as  the  second  of  equations  (3). 

In  like  manner  if  /(a:)  also  has  a  ^--fold  root  equal  to  b,  and  an 
5-fold  root  equal  to  c,  and  so  on,  then 

f(x)^(x—ay(ix—by(x  —  cy  .   .   .   ^(x)',  (10) 

and     f(ix)  =  (x—ay-\x  —  by-\x  —  cy-'  .   .  .  ^,{x).     (11) 

.-.   (ix—ay-\x—by-\x—cy-''  .  .  . 

is  the  G.  C.  D.  of  f(x)  a,ndf\x). 


142  THEORY   OF   EQUATIONS.  [lOl. 

Hence  the  multiple  roots  of  an  equation /(i3?)=  0,  if  there  are 
any,  can  be  detected  by  finding  the  G.  C.  D.  of  f(x)  and  f{x) 
by  the  usual  algebraic  process. 

Likewise  the  common  roots  of  any  two  functions  can  be  ob- 
tained by  finding  the  G.  C.  D.  of  the  two  functions,  and  then 
finding  the  roots  of  this  G.  C.  D. 

Ex.    If  f{x)  =  a^-\-x*  —  lSx'  —  x'-\-iSx-SQ  =  0, 

then  fix)  =  5x*  +  4^^  —  39a;-^  —  2x  +  48. 

The  G.  C.  D.  of  f{x)  and/^(a;)  will  be  found  to  be 

x'-\-x  —  6  =  {x  —  2){x-{-3). 

.-.     fyX)~{x-2nx  +  3nx-i)=0, 

and  the  roots  are  2,  2,  —  3,  — 3,  1. 

EXAMPLES. 
Solve  the  following  equations  by  testing  for  equal  roots : 

1.  x=^  +  11x2 +  24x  — 36  =  0. 

2.  x^  — 2x^  — 15x  +  36  =  0. 

3.  X*  — 7x''  +  9x-'  +  27x  — 54=0. 

4.  X*— llx-^  +  44x^  — 76x  +  48  =  0. 

5.  X*  —  bx'  —  9x'  +  Six  — 108  =  0. 

6.  x^— 15x'H- 10x2 +  60x  — 72  =  0. 

7.  x^  — X*  — 5x3  +  x2  +  8x  +  4  =  0. 

8.  X*  — 2x3— 11x2+ 12x  + 36  =  0. 

9.  X'  — 10x2+ 15x  — 6  =  0. 

10.  X*  — 3x3 -6x2  + 28x  — 24  =  0. 

11.  X-'— 10x^  +  20x2— 15x  +  4  =  0. 

12.  x*+10x'  +  24x2  —  32x  — 128  =  0.     . 

13.  x^  +  19x^+130x3+ 350x2 +  125x  — 625  =  0. 

14.  x''  — 5x5  +  5x*  +  9x''  — 14x2  — 4x  +  8  =  0. 

15.  x^  — 2x*  — 6x^  +  8x2  +  9x+.2  =  0. 

16.  x«  +  7x^  +  4x*  — 58x3— 115x2  — 49x  — 6  =  0. 

17.  xs  — 8x3  +  24x2  — 28x+ 16  =  0. 

18.  x^  — 6x3  — 28x2  — 39x  — 36  =  0. 

19.  What  is  the  condition  that  the  cubic  equation 

x3  +  gx  +  r=0  ^ 

shall  have  a  double  root  ? 

20.  Show  that  in  any  cubic  equation  with  rational  coefficients  a  multiple 
root  must  be  rational. 


102.] 


QUADRATURE. 


143 


Quadrature. 

102.*    Let  y  =J{x)  and  y  =  f  (x)  be  the  equations  of  the 
curves  i^iltf  and -L'if' respectively. 


It  is  required  to  find  the  area  included  between  the  curve  L'Af, 
the  a"-axis,  and  the  ordinates  corresponding  to  ic  =  a  =  OQ,  and 
X  =  b  =  ORj  where  b  >  a.     Let  ^denote  the  area  QA'B'R. 

Divide  the  distance  QR  into  (n  +  1)  equal  parts,  each  equal  to 
h  =  dx.  Draw  ordinates  at  the  points  of  division  and  construct 
rectangles  as  shown  in  the  figure. 

Let 


x. 


a  -{-  h  =:  OQi,     X2  =  a-\-2h,  .   .   .  Xn=  a  +  nh  —  0Q„. 


Then 


and  the  sum  of  the  areas  of  the  (n  -\-l)  rectangles  is 

/i/'(a)+V'(.A)  +  /^/(x,)+  .   .  .   V'(.r„). 
lim 


K 


00 


[h]f'(a)+f'ix,)+f'{a;)+  .   .   .  f(x..)\-\.       (1) 


Now  we  know  by  §  79  that 

fuy.^-    ^'^^  /(^  +  h)  —f(x) 

J  y-'^-h  =  0 J • 

Whence  hf'{x)  +  /i/>  =  f(x  +  h)  —f(x) , 

where  ^  is  a  qu^jutity  which  will  vanish  with  h. 


(2) 
(3) 


144  QUADRATURE.  [102. 

Therefore  we  may  put 

¥'(«)-i-Vo=/(^i)-/(a), 


From  these  equations  we  have  by  addition 

:Ehfix)+^hp  =f(h)  -f(a-).  (4) 

The  second  member  of  (4)  is  independent  of  n,  ^hf'{x)  repre- 
sents the  swm  of  the  areas  of  the  (n  +  1)  rectangles  however  great 
their  number,  and  S/i/>  vanishes  when  n  becomes  infinite. 

.-.     K  =  ^^^^-2f\x)Sx=f{b)-f(,a-)=RB-qA.         (5) 

The  notation  used  to  express  this  is 

K=£r{x)dx  =  Kb)-f{a),  (6) 

where  the  symbol  \  is  an  abbreviation  of  the  word  ''sum,''  and 

means,  in  this  case,  the  summation  of  an  infinite  number  of  infinitessi- 
mal  rectangles. 

Therefore,  in  order  to  find  the  required  area,  we  must  first 
obtain  a  function  which  when  differentiated  will  give  f(x);  then 
substitute  in  this  new  function  j{x)  the  abscissas  of  the  bounding 
ordinates  and  take  the  difference  of  the  results. 

Hence  equation  (6)  may  be  written 

K=£  f(x)dx  =  [f(x)]l  =  Kb)-f(a).  (7) 

In  applying  the  formula  we  must  first  Q.jidf(x)  from  f(x),  i.  e. 
we  must  reverse  the  operation  of  differentiation.  In  this  sense  the 
symbol  J  denotes  an  operation  which  is  the  inverse  of  differentiation. 

This  inverse  process  is  called  Integration. 


102.] 


QUADRATURE. 


145 


If  then  the  symbol  D  be  used  to  denote  differentiation,  the  two 
symbols  f  and  D  neutralize  each  other,  i.  e.  fDf(x)=f(x). 

E.  ST.,  if  Dfix)  =  f\x)  =  4ar'  -  3x2  -I-  4x  -  6, 

then  Jd/(x)=  J/(x)=/(x)=a:*  — x3+2x2  — 6xH-c. 

Hence,  to  integrate  an  integral  function  of  x,  increase  the  exponent  of 
each  power  of  x  by  unity  and  divide  the  coefficient  by  the  increased  ex- 


'  lif(x)  is  the  derivative  of/(.r),  then/(j')  is  called  the  Inte- 
gral ot  f{x).  The  curve  LM  may  be  called  the  Integral  Curve 
with  respect  to  L'M'.  Then  we  may  say  that  the  area  bounded 
by  the  D.  C. ,  the  ^r-axis,  and  two  ordinates  is  numerically  equal 
to  the  difference  of  the  two  corresponding  ordinates  of  the  I.  C. 

If  L'M'  lies  below  the  ^-axis  between  A'  and  B',  the  slope  of 
LM  between  A  and  B  will  be  negative  (§  100) .  Hence  RB  <  QA , 
/.  e.  f(^b)  </(a),  and  the  area  is  negative.  The  rectangles  will  then 
lie  above  the  curve. 

Therefore  the  area  will  be  positive  or  negative  according  as  it  lies 
to  the  right  or  left  of  the  curve  viewed  in  the  direction  of  x  increasing. 
If  L'M'  cuts  the  a:-axis  between  A'  and  J5',  the  formula  gives  the 
excess  (positive  or  negative)  of  the  area  which  lies  to  the  right 
over  that  which  lies  to  the  left. 

Ex.  1 .  Find  the  area  of  the  segment  of 
the  parabola  y-=  4ax  cut  off  by  the  double 
ordinate  through  P(x',  y'). 

Here  y  -  2i  ax^  =f{x). 
.-.    Area 

ONP=  r2y'ax^dx  =  2Va  C' x^dx 

=  2/a  [#  J'=  2v  a  •  §x'^ 

=  ix''2/ax'^  =  §x'2/' 

=  §  rectangle  OBPN. 

.  .    Area  OPQ  =  §  rectangle  ABPQ. 

.  ••  Area  between  AB  and  the  curve  is 
equal  to  ^ABPQ. 

That  is,  the  parabola  trisects  the  rect- 
angle. 

11 


B 

Y                                          ^^ 

/ 

x' 

y' 

O 

A 

N 

Q 

^^^^ 

146 


QUADRATURE. 


[102, 


Ex.  2.    The  curve  y  =  x^  —  3a;^H-2a;  cuts  the  x-axis  in  the  points .(0,  0), 
B(l,0),i)(2,0). 

We  now  have  f{x)  =  x^  —  Sx^  +  2x. 


Y 

/ 

.-.  OAB=i   f(x)dx 

=  C  (x^  —  3x'-\-2x)dx 
-          =[j—'  +  -'  +  -l=^' 

\B                  d/ 

=  (4-8  +  4  +  c)-a  +  c)  = 

~i 

/ 

O 

C 

E              DEF  =  ^j-x^-^x'  +  cj^ 

L 

e. 

DjBF=:(-V-- 

-27  +  9  +  c)-(4-8  +  4  +  c)  =  2i. 

EXAMPLES. 

1.  Find  the  area  included  between  the  curve 

y  =  7^  —  9x''-\-2Sx  —  i5, 
the  X-axis,  and  the  lines  x  =  l,  a;  =  3;  also  x  =  3,  x  =  5;  x  =  1,  x  =  5. 

2.  Find  the  area  included  between' the  curve 

y  =  x-  —  2x  —  8, 

the   a;-axis,    and    the  lines    x  =  —  2,    x  =  4;    also    between    the    curve 
y  =  x'^  —  2x-\-l  and  the  same  lines. 

Find  the  area  between  the  x-axis  and  the  curve 

3.  y=x^  —  Sx''  —  9x  —  21. 

4.  y  =  x^-{-ax. 

5.  y  =  x^  —  ix^  —  2x^-{-12x-{-9. 
Find  the  area  between  the  curves 

6.  2/^  =  4ax    and    x^  =  iay. 

8.  y^  =  x^    and    y^  —  cc^, 

9.  y^  =  x»     and    y'^  =  a^.  Ans. 

10.  2/  =  a^  —  x    and    y  =x, 

11.  y  =  oi^  —  x    and    2/^  =^a;i/2. 


m  —  n 


103.] 


MAXIMA   AND   MINIMA. 


147 


12.  y^  =  ^ax    and    y  =  2x  —  4a. 

13.  x^y  =  a%    x  =  b,    x  =  Cy    and    y  =0,       Ans.    ^^(-r~  )• 

14.  y=x^  —  5x  +  4    and    x  + 1/  =  4. 

15.  2/  =  ar*    and    y  —  x*. 

16.  Show  that  the  area  included  between  the  curve  y  —  Ax",  the  x-aiis 
and  the  line  X  =  a  is  ^  _^,,  where  b  is  the  ordinate  corresponding  to 
x  =  a.    Show  that  the  parabola  is  a  particular  case. 


Maxima  and  Minima. 

1 03.*    Let  the  curves  L3L  L'M'  and  L"M"  be  the  loci,  respect- 
ively, of  the  equations 

y=Kx),  (1) 

y^noc),  (2) 

and  y=r(^0.  (3) 

Then  L"M"  is  the  Second  Derivative  Curve. 

i 
Y  t>   B 


P"'  b" 

Since f"{x)  is  the  first  derivative  oif(x),  the  ordinate  of  L"M" 
at  any  point  represents  the  slope  of  L'M'  at  the  corresponding 
point;  and  the  intersections  E",  F",  G"  of  L"M"  with  the  ic-axis 
correspond  to  the  elbows  E',  F',  G'  of  L'M'  (§  100). 

Let  the  line  x  =  a  meet  the  curves  in  the  corresponding  points 
P,  P',  P',  and  the  a:-axis  in  Q. 


148  MAXIMA   AND   MINIMA.  [103. 

Then  QP=Ka),      QR  =  f(a),      QP-  =  f'{a). 

That  is,  QP'  is  the  slope  of  LM  at  P,  and  QP"  is  the  slope  of 
Jj'M'sitP'. 

Suppose  the  point  P  to  move  along  the  curve  LM  toward  the  right. 
As  P  approaches  the  elbow  B,  the  ordinate  QP  increases ;  but  as 
JP  passes  through  B,  the  ordinate  eeases  to  increase  and  begins  to 
decrease.  At  such  a  point  the  ordinate,  i.  e.  f(x)j  is  said  to  have  a 
Maximum  Value,  or  to  be  a  Maximum*.  In  like  manner  as 
_P  approaches  the  elbow  J.,  or  C,  the  ordinate  §P  decreases;  but 
a-s  P  passes  through  ^,  or  C,  the  ordinate  ceases  to  decrease  and 
begins  to  increase.  At  such  points  QP,  i.e.  f(x),  is  said  to  have  a 
MinimurQ  Value,  or  to  be  a  Minimum. 

That  is,  a  function,  f(x),  is  said  to  have  a  maximum  value  when 
ijc  =  a,  if  f(cf')^  f(cL  zh  h);  and  a,  minimum  value,  if /(«)</(«  ±  h), 
for  very  small  values  of  h. 

Since  in  these  definitions  the  comparison  is  made  between  values 
of  f{x)  in  the  immediate  vicinity  only  oi  A,  B,  0,  a  maximum  is 
not  necessarily  the  greatest,  nor  a  minimum  the  least,  of  all  the 
-values  of  the  function. 

Moreover,  since  maximum  and  minimum  ordinates  occur  only 
^t  the  elbows  of  a  curve  where  the  tangent  is  parallel  to  the  x-Skxis, 
^  necessary  but  not  a  sufficient  condition  for  a  maximum  or  mini- 
mum value  oif(x)  is/'(^)=:0  (§  100). 

Suppose  a  tangent  to  be  drawn  to  L3I  at  any  elbow,  i.  e.  at  any 
point  where /'(ir)=  0.  Then  the  curve  will  lie  below  or  above 
this  tangent  line  for  a  short  distance  on  both  sides  of  the  elbow, 
according  as  the  ordinate  of  the  elbow  is  a  maximum  or  a  mini- 
mum. If  the  curve  crosses  this  tangent,  as  at  D,  the  ordinate  is 
xeither  a  maximum  nor  a  minimum. 

Hence,  as  P  passes  (toward  the  right)  through  an  elbow,  as  B, 
"whose  ordinate  is  a  maximum,  the  slope  of  LM,  i.  e.  f(x), 
•changes  irom. positive  to  negative;  and  as  P  passes  through  an  elbow , 
fiuch  as  A  or  C,  whose  ordinate  is  a  minimum /(a^)  changes  from 
■negative  to  positive. 

Therefore,  the  necessary  and  sufficient  conditions  that  S{x) 
«!hall  be  a  maximum  or  a  minimum  when  x  =  a  are  as  follows: 

JFor  max.,f(a)  =  0;  f(a—h),  positive;  f(a  +  h),  negative.  1 
For  min. ,  f(a)  =  0 ;  /(«  —  h),  negative;  f(a  -\-  h),  positive.  J 


103.]'  MAXIMA    AND    MINIMA.  149 

If  /(a  +  A)  and /'(a  —  h)  have  the  same  sign,  /(a)  is  neither  a 
maximum  nor  a  minimum  value  of  f(x). 

Now  suppose,  as  is  usually  the  case,  that  a  is  a  single  root  of 
/(ic)  =0,  so  that /'(a)  ^  0.     (§  101.) 

Then  if  QP  passes  through  a  maximum  value,  as  B'B^  when 
x^a,  the  slope  of  LM  changes  from  -j-  to  — .  Hence  the  corre- 
sponding point  F'  crosses  the  ic-axis  from  above  downwards,  and 
therefore  the  slope  of  L'M'  at  B'  is  negative,  i.  e. 

B'B"=f\a)  is  negative. 

If  QP  passes  through  a  minimum  value,  as  C  C,  the  slope  of 
LM  changes  from  —  to  +.  Hence  P'  crosses  the  ^r-axis  from 
below  upwards,  and  therefore  the  slope  of  L'M'  at  C  is  positive,  i.  e, 

C  C"=f"{a)  is  positive. 

Therefore,  if  /"(a)  7^  0,  the  necessary  and  sufficient  conditions 
that /(a)  shall  be  a  maximum  or  a  minimum  value  of  f{x)  are: 

For  a  maximum,    f(a)  =  0;    fia),  negative. 
For  a  minimum,    /'(a)=0;    f\a),  positive 

If  a  is  an  r-fold  root  of  f(x)  =  0,  then  /'(a)  =  0  when  r  >  1 
(§  101)  and  the  conditions  (5)  fail  to  disclose  the  nature  of  the 
corresponding  ordinate. 

If  r  is  an  odd  number  the  curve  L'M'  will  cut  the  x-a>x[s  in  an 
odd  number  of  coincident  points,  and  hence  will  cross  the  a?-axis 
at  the  point  (a,  0).  Therefore  the  sign  of  f(x)  will  change  from 
+  to  —  for  a  maximum,  and  from  —  to  +  for  a  minimum.  In 
this  case  we  must  use  conditions  (4)  to  determine  the  nature  of 

If  r  is  an  even  number,  L'M'  will  not  cross  the  a:-axis  at  the 
point  (a,  0),  as  at  D'.  Hence  f(x)  will  not  change  sign,  and 
therefore /(a)  is  neither  a  maximum  nor  a  minimum. 

The  maximum  and  minimum  ordinates  of  L'M'  can  be  deter- 
mined in  the  same  manner.  The  points  E,  F,  G,  D  on  LM  cor- 
responding to  the  maximum  and  minimum  ordinates  of  L'M  are 
therefore,  respectively,  the  points  of  maximum  and  minimum 
slope  of  LM.  At  the  points  where  the  slope  of  a  curve  ceases  to 
increase  and  begins  to  decrease,  or  vice  versa,  the  curve  changes 
the  direction  of  its  curvature.  Therefore  E,  F,  G,  D  are  the 
points  of  inflection  of  LM  (§  100). 


native.  I 
itive.  J 


150 


MAXIMA   AND    MINIMA. 


[103. 


Hence  the  position  of  the  points  of  inflection  of  a  curve  are  obtained 
by  finding  the  position  of  the  maximum  and  minimum  ordinates  of  the 

D.a 


104.     Illustrative  Examples. 

Ex.  1.    The  curves  y  =  sin  x  and  y  —  cos  x  are  good  examples  of  the 
relations  and  principles  explained  in  §  102  and  §  103. 


Let 

^,  ^,,  ^       lim   sin(a; 

Then    fix)  =  j,^Q-^- 


f{x)  =  sin  X. 
h)  —  sinx       lim 


lim    r       ,     ,   ,,  ,sin  m~\ 
-/i  =  OL^os(x  +  i/i)-^J        (1) 

(Ex.  13,  p.  115.)    (2) 


=  cos  X. 
In  like  manner  it  can  be  shown  that  the  derivative  of  cos  x  is 


Let 


and 


y  —f{x)  =  sin  X,    equation  of  LM, 
y  —f\x)  =  cos  X,    equation  of  L^M^, 
y  =f{x)=  —  sin  X,    equation  of  L^^M^\ 


Then  f^x)  =  cos  x  =  0,    when    x  =  ^tt,    |-,     ^^k^  etc. 

and  /'^(i^)  =  —  sin  ^a  =  —  1.  .-.    sin  j^tt  =  1  is  a  max. 


r'(l^) 


1. 


sin  irr  =; 


1  isamin.,etc. 


Also 


f^^{x)  =  —  sin  X  =  0,    when    x  =  tt,    2t,    3-,  etc. 


These  values  of  x  make  cos  x  alternately  a  maximum  and  a  minimum, 
and  hence  give  the  points  of  inflection  of  LM.  That  is,  the  sine  curve 
changes  the  direction  of  its  curvature  as  it  crosses  the  a;-axis. 

Let  X  =  OQ  be  any  line  parallel  to  the  y-axis. 

Thenfix)  =  cos  x  =QP'=  slope  of  LM  at  P. 

Moreover,  by  ^  102  we  have 


Area  OAP^Q  =  I   f\x)dx  =  |    cos  xdx  =   sin  a;     =Bmx  =  QP. 


(3) 


That  is,  the  ordinate  of  any  point  of  the  cosine  curve  is  equal  to  the  slope 
of  the  sine  curve  at  the  corresponding  point;  and  the  ordinate  of  the  sine 


104.] 


MAXIMA    AND    MINIMA. 


151 


curve  is  equal  to  the  area  bounded  by  the  ordinate,  the  cosine  curve,  and 
the  axes  of  coordinates. 

Ex.  2.    Find  the  maximum  and  minimum  values  of  the  function 

Y 

fix)  -x*  —  ix^  —2x'  +  12a;  +  4. 
Here  f{x)  =  ix'  —  \.2x'  —  4x  + 12 
and      f  '{X)  =  12a;2  —  24a;  —  4. 

The  roots  of /'(x)  =  0 
are  —  1,    1,    3. 

/'^(-1)=32. 
///(!)  =  _  16. 
/'^(3)=32. 


/( —  1)  =  —  5  is  a  minimum. 
/(I)  =  11  is  a  maximum.  » 
/(3)  =  —  5  is  a  minimum. 


The  roots  of  f^\x)  =  0  are  1  ±  f  i/3,  which  are  the  distances  of  the  points 
of  inflection  from  the  y-Sixis. 

In  the  solution  of  problems  in  maxima  and  minima,  we  must  first  obtain 
an  algebraic  expression, /(x),  for  the  quantity  whose  maximum  or  minimum 
is  required.     We  may  then  proceed  as  in  the  preceding  examples. 

Ex.  3.  Find  the  maximum  rectangle  that  can  be  inscribed  in  a  given 
triangle. 

Let  b  =  the  base  of  the  given 
triangle  ABC,  h  the  altitude  and 
X  the  altitude  of  the  inscribed 
rectangle.  Then  from  similar  tri- 
angles, 

EG.b=(h  —  x)  -.h. 


EG^-Ah  —  x). 


Then  -i^ihx  —  x-)  is  the  area 


of 


the  rectangle,  which  is  to  be  made 
a  maximum.    Any  value  of  x  that  will  make  {hx  —  x^)  a  maximum  will  also 

make  -r{^x  —  x^)  a  maximum.    Hence  we  may  put 


Then 
Also 


f{x)  =  hx  —  x^. 
f\x)=h  —  2x  =  0    yfhenx  =  ih. 
r'ix)  =  -2. 
.'.    f[\h)=\h^  is  a  maximum. 


Therefore  the  altitude  of  the  maximum  inscribed  rectangle  is  one-half 
the  altitude  of  the  triangle. 


152 


MAXIMA   AND    MINIMA. 


[104. 


Ex.  4.    Find  the  area  of  the  largest  rectangle  which  can  be  inscribed  in 
the  ellipse 


2/^_ 


a^  +  &  =  ^- 


(1> 


Let  K  denote  the  area  of  the 
rectangle.    Then 
46 


K=2x-2y=^V  a^x'^  —  x'   (2> 

is  the  function  of  x  which  is  to  be 
a  maximum. 

Any  value  of  x  which  will  make 
ax"^  —  a;*  a  maximum,  or  a  minimum, 
will  also  make  K  a  maximum,  or  a 
minimum. 
Therefore,  let  f(x)  =  ax^  —  x*. 

Then  f\x)=  2a'x  —  ix^  =  0    when  x  =  0,  or  ±  ^0/2,, 

and  f''{x)=2a^—12x-'=—ia'    when  x  =  ^0^/2. 

.*.    x  =  hai/2  will  make  K  a  maximum. 

Therefore  K  =  2ab  is  the  area  of  the  maximum  rectangle,  which  is  half 
the  rectangle  whose  sides  are  the  axes  of  the  ellipse. 

Ex.  5.    Find  the  dimensions  of  a  cone  of  revolution  which  shall  have  the 
greatest  volume  with  a  given  surface. 


Let  X  =  the  radius  of  the  base, 
S  —  the  total  surface. 


Then 
and 


S  =  rcx^  -J-  TTxy ;  whence  y 
{Altitude^  =  2/'  —  x^  =  -_^  ^ 


the  slant  height,  F=  the  volume,  and 


-X  ' 

S'         2S 


Let 
Then 


and 


2S      yS'x'  —  27rSx* 


3 


fix)  =  Sx-"  —  2-x*. 


1     iS 


r{x)=  2Sx  —  87rx3=  0    when  x  =  0,  or  ±  ^y^^ 

1     IS 


f''{x)=  2S—  24rrx2=  —  is    when  x  = 


n'-^' 


F  IS  a  max.  when  x  =  -sV— ,  and  2/  =  «  A/—* 


That  is,  if  the  surface  is  constant,  the  volume  of  the  cone  is  a  maximum 
when  the  slant  height  is  three  times  the  radius  of  the  base. 


104.]  MAXIMA    AND    MINIMA.  153 


EXAMPLES. 

Find  the  maximum  and  minimum  ordinates  and  the  points  of  inflection 
(points  of  maximum  or  minimum  slope)  of  the  curves 

3.    2/  =  x3  — 3|c2  4-6xH-7..  f),    2/^x-''  — 9x2  +  24x  +  16. 

5.  Find  the  sides  of  the  maximum  rectangle  which  can  be  inscribed  in 
a  circle;     in  a  semi-circle. 

6.  Find  the  sides  of  the  maximum  rectangle  which  can  be  inscribed  in 
a  semi-ellipse. 

7.  Find  the  altitude  of  the  maximum  rectangle  which  can  be  inscribed 
in  a  segment  of  a  parabola,  the  base  of  the  segment  being  perpendicular 
to  the  axis  of  the  parabola. 

8.  What  is  the  least  square  that  can  be  inscribed  in  a  given  square  ? 

9.  Find  the  altitude  of  a  cylinder  inscribed  in  a  cone  when  the  volume 
of  the  cylinder  is  a  maximum. 

/ 1^  What  are  the  most  economical  proportions  for  a  cylindrical  tin  can  ? 
That  is,  what  should  be  the  ratio  of  the  height  to  the  radius  of  the  base 
that  the  capacity  shall  be  a  maximum  for  a  given  amount  of  tin? 

1.)   What  are  the  most  economical  proportions  for  a  cylindrical  tin  cup  ? 


■^ 


12.  What  are  the  most  economical  proportions  for  an  open  cylindrical 
water  tank  made  of  iron  plates,  if  the  cost  of  the  sides  per  square  foot  is 
two -thirds  of  the  cost  of  the  bottom  per  square  foot  ? 


t3j/  An  open  box  is  to  be  made  from  a  sheet  of  pasteboard  12  inches 
square  by  cutting  equal  squares  from  the  four  comers  and  bending  up  the 
sides.    What  are  the  dimensions  of  the  largest  box  that  can  be  made? 

/14j/  If  a  rectangular  piece  of  pasteboard,  whose  sides  are  a  and  6,  have 
a  s^are  cut  from  each  corner,  find  the  side  of  the  square  so  that  the  re- 
maiijder  may  form  a  box  of  maximum  capacity. 

1^/  A  person  being  in  a  boat  3  miles  from  the  nearest  point  of  the  shore, 
wishes  to  reach  in  the  shortest  possible  time  a  place  5  miles  from  that  point 
along  the  shore ;  supposing  he  can  walk  5  miles  an  hour,  but  can  row  only 
at  the  rate  of  4  miles  an  hour,  required  the  place  where  he  must  land. 

16.  The  cost  per  hour  of  driving  a  steamer  through  still  water  varies  as 
the  cube  of  its  speed.  At  what  rate  should  it  be  run  to  make  a  trip  against 
a  four-mile  current  most  economically  ? 

17.  Find  the  altitude  of  the  greatest  cylinder  that  can  be  cut  out  of  a 
given  sphere. 


154:  MAXIMA    AND    MINIMA.  [104. 

18.  Find  the  altitude  of  the  greatest  cone  that  can  be  inscribed  in  a 
given  sphere. 

19.  Find  the  altitude  of  a  cone  inscribed  in  a  sphere  which  shall  m^ke 
the  convex  surface  of  the  cone  a  maximum. 

20.  Find  the  dimensions  of  a  cone  with  a  given  convex  surface  and  a 
maximum  volume. 

21.  Find  the  altitude  of  the  least  cone  that  can  be  circumscribed  about 
a  given  sphere. 

22.  Find  the  altitude  of  the  maximum  cylinder  that  can  be  inscribed  in 
a  given  paraboloid. 

23.  What  is  the  diameter  of  a  ball  which,  being  let  fall  into  a  conical 
glass  of  water,  shall  expel  the  most  water  possible  from  the  glass;  the 
depth  of  the  glass  being  6  inches  and  its  diameter  at  the  top  5  inches  ? 

Ans.    m  in. 

24.  The  sides  of  a  rectangle  are  a  and  b.  Show  that  the  greatest  rect- 
angle that  can  be  drawn  so  as  to  have  its  sides  passing  through  the  comers 

of  the  given  rectangle  is  a  square  whose  side  is  — ^t__ . 

25.  The  strength  of  a  beam  of  rectangular  cross -section,  if  supported 
at  the  ends  and  loaded  in  the  middle,  varies  as  the  product  of  the  breadth 
of  the  cross-section  by  the  square  of  its  depth.  Find  the  dimensions  of 
the  cross-section  of  the  strongest  beam  that  can  be  cut  from  a  log  18  inches 
in  diameter. 

26.  A  Norman  window  consists  of  a  rectangle  surmounted  by  a  semi- 
circle. If  the  perimeter  of  the  window  is  given,  show  that  the  quantity  of 
light  admitted  is  a  maximum  when  the  radius  of  the  semicircle  is  equal  to 
the  height  of  the  rectangle. 


CHAPTER  VII. 
OONIO  SECTIONS. 

105.  The  general  equation  of  the  first  degree  and  also  some 
special  cases  of  the  equation  of  the  second  degree  have  been  con- 
sidered in  Chapter  III.  We  now  proceed  to  the  study  of  the 
general  equation  of  the  second  degree,  and  the  standard  forms  to 
which  it  can  be  transformed.  It  will  presently  be  shown  that 
the  locus  of  such  an  equation  is  always  a  curve  that  can  be  ob- 
tained by  making  a  plane  section  of  a  right  circular  cone.  For 
this  reason  the  locus  is  called  a  Conic  Section.* 

106.  The  Fundamental  Property  of  a  Plane  Section  of  a  Bight 
Circular  Cone,  or  a  Conic  Section. 

Let  VO  be  the  axis  of  a  right  circular  cone,  and  APB  any  sec- 
tion made  by  a  plane  not  passing  through  V. 

Inscribe  a  sphere  in  the  cone  tangent  to  the  plane  of  the  section 
at  F;  then  the  line  of  contact  HRK  ci  the  sphere  and  cone  is  a 
circle  with  centre  C  in  VO,  whose  plane  is  perpendicular  to  VO 
and  meets  the  plane  of  the  section  APB  in  the  line  ES. 

Pass  the  plane  VMN  through  VO  perpendicular  to  the  plane 
APB,  meeting  it  in  the  line  AB,  meeting  the  plane  HKR  in  HK, 
and  the  line  ES  in  D ;  then  the  plane  VMN  is  also  perpendicular 
to  the  plane  HKR,  and  therefore  perpendicular  to  ES. 

Let  P  be  any  point  on  the  section. 

♦After  studying  the  straight  line  and  the  circle,  the  old  Greek  mathematicians  turned 
their  attention  to  the  conic  sections,  and  by  investigating  them  as  sections  of  a  cone  soon 
discovered  many  of  their  characteristic  properties.  The  most  important  of  these  discov- 
eries were  probably  made  by  Archimedes  and  ApoUonius,  as  the  latter  wrote  a  treatise  on 
conic  sections  about  200  B.  C- 

These  curves  are  worthy  of  careful  study,  not  only  on  account  of  their  historic  inter- 
est, but  also  on  account  of  their  importance  in  the  physical  sciences  and  their  frequent 
occurrence  in  the  experiences  of  everyday  life.  For  example,  the  orbit  of  a  heavenly 
body  is  a  conic  section.  For  this  reason  they  were  thoroughly  studied  by  the  astronomer, 
Kepler,  about  1600  A.  D.  The  path  of  a  projectile  is  a  parabola.  The  law  of  falling  bodies, 
the  pressure- volume  law  of  gases,  the  law  of  moments  in  uniformly  loaded  beams,  all 
give  conic  sections.  The  bounding  line  of  a  beam  of  uniform  strength,  the  oblique  sec- 
tion of  a  stove-pipe,  the  shadow  of  a  circle,  the  apparent  line  dividing  the  darlc  and  light 
parts  of  the  moon,  etc.,  are  all  conic  sections.  The  reflectors  in  head-lights  and  search- 
lights are  parabolic. 


156 


CONIC   SECTIONS. 


[106. 


M' 


11' 
II' 

KL^. 

LV 

1    X          X^*" 

—     Tk' 

B 


Draw  PF,  and  the  element  PV  which  will  be  tangent  to  the 
sphere  at  i?. 


106.]  CONIC   SECTIONS.  157 

Through  P  draw  a  line  perpendicular  to  the  plane  HKR,  which 
will  meet  CR  produced  in  Q ;  and  through  PQ  pass  a  plane  per- 
pendicular to  ES  meeting  it  in  S. 

Let  /5  =  Z  PRQ  =  Z  AHD,  the  complement  of  the  semi- vertical 
angle  of  the  cone. 

Let  a  =  iADH=  IPSQ. 

Then,  since  tangents  from  an  external  point  to  a  sphere  are 
equal, 

PF  =  PR. 

From  the  right  triangles  PQR  and  PSQ  we  get 

PQ  =  PR  sin  ^  =  PS  sin  a. 

•  •     PS      sm,3'  ^^^ 

So  long  as  we  consider  any  particular  section  the  point  F  and 
the  line  ES  are  fixed,  a  is  constant,  and  therefore  the  ratio  of  PF 
to  PS  is  constant. 

Equation  (1)  expresses  the  Fundamental  Property  of  a  Conic  Sec- 
tion, which  is  used  as  the  defining  property.  Moreover,  all  curves 
which  have  this  property  are  plane  sections  of  some  cone;  for 
all  possible  curves  satisfying  this  condition  are  gotten  by  giving 
this  constant  ratio  all  possible  values,  and  also  letting  the  dis- 
tance, FDy  from  the  fixed  point  to  the  fixed  line  have  all  possible 
values.  We  can  do  this  with  a  conic  section.  For  any  particu- 
lar value  of  /?,  i.  e.  for  any  particular  cone,  the  ratio  can  vary 
from  zero  (when  a  =  0)  to  esc  /5  (when  a  =  Jtt).  For  any  particu- 
lar value  of  a  the  ratio  can  vary  from  sin  a  (when  /5  =:  ^n^)  to  oo 
(when  /5  =  0).  Thus  the  ratio  can  have  any  value  from  0  to  oo  . 
Also  the  distance  of  Firom.ES,  depending  as  it  does  upon  the 
size  of  the  inscribed  sphere,  for  any  particular  cone  and  any  par- 
ticular value  of  a  can  vary  from  zero  to  oo  ♦.  Therefore  the  prop- 
erty expressed  by  (1)  is  indeed  a  defining  property  of  a  conic 
section,  that  is: 

A  Conic  Section,  or  A  Conic,  is  the  locus  of  a  point. which  moves  in  a 

plane  so  that  its  distance  from  a  fixed  point  in  the  plane  is  in  a  constant 

ratio  to  its  distance  from  a  fixed  line  in  the  plane.  * 

♦This  is  generally  known  as  Boscovich's  definition  of  a  conic  section,  but,  in  the  ar- 
ticle on  Analytic  Geometry  in  the  Encyclopedia  Britannica,  ninth  edition,  Cayley  calls  it 
the  definition  of  Apollonius. 


158  CONIC   SECTIONS.  [107. 

The  fixed  point  F  is  called  the  Focus ;  the  fixed  line  ES  is 
called  the  Directrix ;  the  constant  ratio  is  called  the  Eccentric- 
ity, and  is  denoted  by  the  letter  e. 

107.     Classifieation  of  the  Conie  Sections. 

Using  e  to  denote  the  eccentricity,  we  have,  by  (1)  of  §  106, 
PF_  sing  _ 
FS~  sin  13-^-  (^> 

When  a  <  y5,  e  <  1 ;  the  plane  of  the  section  meets  all  the  ele- 
ments of  the  cone  on  the  same  side  of  the  vertex ;  the  section  is  a 
closed  curve  as  shown  in  the  figure  §  106,  and  is  called  an  Ellipse. 

When  a  =  0,  e  =  0 ;  the  plane  of  the  section  is  perpendicular  to 
the  axis  of  the  cone,  VO,  and  the  section  is  a  Circle.  Hence  a 
circle  is  a  particular  case  of  the  ellipse. 

When  a  =  I3,e=l;  the  line  AB  (§  106)  is  then  parallel  to  VJS 
and  the  point  B  moves  off  to  an  infinite  distance ;  the  section 
c  /nsists  of  a  single  branch  extending  to  infinity,  and  is  called  a 
Parabola. 

When  a  >  /?,  e  >  1  and  the  plane  APB  (§  106)  meets  iVF pro- 
duced on  the  other  sheet  of  the  conical  surface;  the  section  is 
then  composed  of  two  infinite  branches,  one  lying  on  each  sheet 
of  the  cone,  and  is  called  a  Hyperbola. 

Thus  the  parabola  is  the  limiting  case  of  both  the  ellipse  and 
the  hyperbola. 

Let  the  plane  of  the  section  pass  through  the  vertex  of  the  cone. 

Then  if  e  <  1,  the  section  is  a  point  ellipse  or  a  point  circle. 

If  e  =  1,  the  plane  is  tangent  to  the  cone  and  the  parabola  re- 
duces to  two  coincident  straight  lines. 

If  e  >  1 ,  the  hyperbola  becomes  two  intersecting  straight  lines, 
which  approach  in  the  limit  two  parallel  lines  as  the  vertex  of  the 
cone  moves  off  to  an  infinite  distance. 

Hence  a  point,  two  intersecting  straight  lines,  two  parallel 
straight  lines,  and  two  coincident  straight  lines  are  all  limiting 
cases  of  conic  sections. 

Under  the  head  of  conic  sections  we  must  therefore  include : 

(1)  The  Ellipse,  including  the  circle  and  the  point; 

(2)  The  Parabola; 

•        (3)    The  Hyperbola; 
(4)    The  Line-pair. 


108.] 


CONIC   SECTIONS. 
EXAMPLES. 


169 


1.  Inscribe  a  sphere*  tangent  to  the  plane  APB  (fig.  §  106)  on  the  other 
side  and  thus  show  that  the  ellipse  has  another  focus  and  a  corresponding 
directrix;  and  that  the  two  directrices  are  parallel  and  equidistant  from 
the  foci. 

2.  By  means  of  these  two  inscribed  spheres,  prove  the  property  of  the 
ellipse  given  in  §  34. 

3.  Inscribe  spheres*  in  both  sheets  of  the  cone  and  show  that  the  hyper- 
bola also  has  two  foci  and  two  directrices. 

4.  Prove  the  property  of  the  hyperbola  stated  in  §  36. 

5.  Where  are  the  foci  and  the  directrices  of  the  circle,  the  parabola, 
and  two  intersecting  straight  lines? 


General  Equation  of  the  Conic  Sections. 

108.     To  find  the  equation  of  a  conic  section  in  rectangular  coordi- 
nates. 


Let  the  equation  of  the  directrix  EC  b3 

X  cos  o.  -\-  y  sin  a  —  p  =  0. 

Let  F{k,  V)  be  the  corresponding  focus. 

Let  P{,ic^  y)  be  any  point  on  the  conic. 

Draw  PS  perpendicular  to  EC,  and  join  P  and  F. 

Then  from  equation  (1)  of  §  107  we  have 

PF=e'PS. 


a) 


(2) 


*  For  complete  diagrams  see  "  Some  Mathematical  Curves  and  Their  Graphical  Con- 
struction," by  F.  N.  Wilson,  pp.  45,  46. 


160  CONIC   SECTIONS.  [109. 

Now  ;         PF^=:^x-ky^{y-l)\  [(2),  §7] 

and  PS  =  X  cos  a  -\-  y  sin  a  — p.        [(4),  §  50] 

Therefore  the  required  equation  is 

(x  —  ky-\-  {y  —  iy=  e\x  cos  a  -\-  y  sin  a^py.  (3) 

Expanding  (3)  and  collecting  terms  we  have 

(1  —  e-cos^a)a^  —  2(e^sinacosa)^i/  +  (1  —  e^sin^«)?/^ 

H-  2{ey  cos  a  —  k)x-{-  2(^e^p  sin  a  —  l)y-{-  k'-{-  f  —  ey=  0.    (4) 

Since  equation  (4)  contains  five  arbitrary  constants,  k,  I,  a,p,  e, 
it  may  be  any  equation  of  the  second  degree.  That  is,  any  equa- 
tion of  the  second  degree  represents  a  conic  section. 

The  most  general  equation  of  a  conic  is,  therefore,  the  complete 
equation  of  the  second  degree,  and  may  be  written  (§  53) 

ax'+  2hxy  +  6?/'+  2gx  +  2/z/  H-  c  =  0.  (5) 

Equations  (4)  and  (5)  each  contain  five  arbitrary  constants. 
A  conic  section  can  therefore  be  made  to  satisfy  five  independent 
conditions,  and  no  more.  That  is,  a  conic  can  be  made  to  pass 
through  any  five  given  points. 

If  the  directrix  EC  be  taken  for  the  ?/-axis,  and  FD,  perpen- 
dicular to  EC,  for  a?-axis,  the  equation  of  the  conic  (3)  reduces  to 

{x  —  ky^y'=e'xK  (6) 

1 09.  To  find  the  parameters  of  the  conic  represented  by  the  general 
equation 

ax'+  2hxy  +  by'-\-  2gx -]-2fy -{- c  =  0.  ( 1 ) 

The  equation  referred  to  parallel  axes  through  the  point 
{x',  y')  will  be  found  by  substituting  x -\-  x'  for  x  and  y  -j-  y'  for 
y  [§  66,  (10)] ,  and  will  therefore  be 

a(ix  +  xy+  2h{x  +  x'){y  +  y')-^  b(^y  +  y^ 

+  2g(ix  +  ^')  +  2/(2/  +  y )  +  c  =  0, 

or     ax'^2hxy^by'+2x(iax/-^hy'-^g)-{-2y(hx'+by'-^f) 

-h  ax"-\-  2hx'y'-^  by""^  2gx' -^  2fy'^  c  =  0.      (2) 

If,  as  is  generally  possible,  x'  and  y'  be  so  chosen  that 

a^'-f%'+5r=:0,  (3) 

and  hx'-\-by'^S^%  (4) 


109.]  CONIC  SECTIONS.  161 

the  coefficients  of  x  and  y  in.  (2)  will  both  vanish,  and  the  equa- 
tion referred  to  (a?',  y')  as  origin  will  then  be 

a^'-f  2hxy  +  hif-\-  c'  =  0,*  (5) 

where  &  =  ax''-\-  2hx'y'+  by''-\-  2gx'+  2fy'+  c.  (6) 

The  locus  represented  by  (5)  is  symmetrical  with  respect  to 
the  origin  [§  28,  (9)] ;  i.  e.  all  chords  which  pass  through  the 
origin  are  bisected  at  the  origin. 

The  point  (x',y')is  therefore  called  the  Centre  of  the  Conic. 

Hence  the  coordinates  of  the  centre  of  the  conic  represented  by 
(1)  are  the  values  of  x'  and  y'  which  satisfy  equations  (3)  and  (4), 

'"^'  ""-ah  —  h?'         y-ab—h''  ^^^ 

Multiply  equations  (3)  and  (4)  by  x'  and  y',  respectively,  and 
subtract  the  sum  from  the  right  member  of  (6)  ;  we  thus  get 

e'=gx'-^fy'JrC.  (8) 

_  abe  +  2fgh  -  af  -  bg'  -  cW  _      A  .....      .q^ 

Suppose  equation  (4)  of  §  108  and  equation  (5)  to  represent  the 
same  locus;  then  their  coefficients  must  be  proportional,  and  we 
have  the  following  equations  for  determining  the  parameters  of 
the  conic  represented  by  equation  (5). 


1  —  e^  cos^ 

a      1  —  e^  sin^  a      —  e^  sin  a  cos  a      e^p  cos  a 

—  h 

a 

~           b           ~              h                            0 

' 

e'psina  —  l      k'-^P  —  ey 

=     0      "      c      -^^^y- 

(10) 

Then 

l—e^cos'a      1  — e^sin^a      2  — e^ 

(11) 

'*"           a          ~~           b           ~a-\-b' 

7^  = 

1  —  e^  -\-e*  sin^  a  cos^  a      e*  sin^  a  cos^  a       1  — 

(12) 

ab                   ~~           h'           ^ab- 

*  Observe  that  the  coefficients  a,  6,  and  h  are  not  changed  in  this  transformation,  and 
therefore  the  following  equations  which  involve  only  o,  6,  and  h  are  the  same  for  (1)  and  (5) . 

12 


162 


CONIC   SECTIONS. 


/2  — eV      l  —  e' 

'     \a-i-  b)  ~  ab  —  h' 


[109. 


(13) 


Whence       (a6  —  h')e'-\-  [(a  —  6)^+  W] {e'  —  1)=  0.  (14) 

.    .     1 


Considering  (14)  as  a  quadratic  in  -^,  and  solving  gives 
1       1  a-f6 


Also         T  = 


e'      2  -2l/(a— 6)^+4/1^* 
3^  sin  a  COS  a  6^  sin  2a      2  —  e^ 


^  2h  a^b' 

2h    /2 


(15) 

(16) 


a  +  6\e^         / 


sin  za  = 


2^ 


Whence 


cos  2a  = 


and 


tan  2a 


V{a—by-{-4:h' 

a  —  b 

l/(a— 6)2+4p"' 

2A 


.  by  (15) 


^        (17) 


From  (10)  and  (16)  we  get 

_  F  H-  f  —  ey 


^  sin  2a 
2h     • 


(18) 


Since  the  denominators  of  two  fractions  in  (10)  are  zero,  their 
numerators  must  also  be  zero. 


,•.     Ic  —  &'p  cos  a,         I  —  e^p  sin  a. 
Squaring  and  adding  (19)  gives 

Substituting  (20)  in  (18)  we  have,  from  (9), 
2        c'  sin  2a  l\  sin  2a 


2^(1  —  e^)      2h(l  —  e')(ab  —  h'y 


Then 


k'-{-P 


c'V^a  —  bYi-W 


ab  —  K' 


(19) 
(20) 

(21) 
(22) 


The  value  of  all  the  parameters  can  now  be  found.  Thus 
equations  (15)  and  (17)  give  the  values  of  e  and  a  respectively; 
then  the  value  of  p  can  be  found  from  (21)  ;  and  lastly,  the  values 


109.]  CONIC   SECTIONS.  163 

of  k  and  I  can  be  obtained  from  (19).  It  must  be  borne  in  mind 
that  the  values  of  h,  I,  and  p  given  by  (19),  (20),  and  (21)  are 
measured  from  the  centre  of  the  conic. 

For  any  particular  value  of  e,  equation  (21)  gives  two  values 
of  p  which  differ  only  in  sign.  Substituting  these  two  values  of 
|)  in  (19)  we  get  two  pairs  of  values  of  k  and  I  which  differ  only 
in  sign.  Hence  every  conic  has  two  directrices  which  are  parallel 
and  equidistant  from  the  centre,  and  two  corresponding  foci  which 
are  symmetrical  with  respect  to  the  centre.  Moreover,  since 
from  (19) 

I  =  k  tan  a, 

the  two  foci  lie  on  the  line,  y  =  x  tan  a,  passing  through  the  centre 
perpendicular  to  the  directrices. 

This  line  is  called  the  Principal  Axis  of  the  conic  section, 
and  its  equation  is 

2/=a;tana,     or     y  —  y'=t2ina{x  —  x'),  (23) 

according  as  the  centre  is  at  the  point  (0,  0)  or  {x',  y'). 

Equation  (20)  shows  that  the  distance  from  the  centre  to  the 
foci  is  greater  or  less  than  p  according  as  e  is  greater  or  less  than 
unity.  That  is,  in  the  ellipse  the  foci  lie  between  the  centre  and 
the  directrices,  while  in  the  hyperbola  the  directrices  pass  be- 
tween the  centre  and  the  foci. 

The  Parabola.     When  e  =  1,  we  have,  from  (14),  ah  — A^=  0. 

Hence  the  coordinates  of  the  centre,  equation  (7),  are  both  in- 
finite; and  therefore  when  (1)  represents  a  parabola  the  transfor- 
mation from  (1)  to  (5)  becomes  impossible.  In  this  case  we  may 
obtain  the  equations  for  the  determination  of  the  parameters  by 
putting  e  =  1  in  equation  (4)  of  §  108,  and  comparing  the  result- 
ing coelB&cients  with  those  of  (1).     This  gives 

sin^  a  _  cos^  a  _  —  sin  a  cos  OL  _p  cos  a  —  k 
a  b  h  g 

^psina-l^k'-i-P-p'__^ 

f  c 

Then  r  = =  — f —  =  — r-i-  (25) 

a  b  a  -\-  b  . 

■'•     '?"=>l^'      •^'"=V«rTft'     tona=^^.     (26) 


164                                                     CONIC   SECTIONS. 

[110. 

Also 

,  _i>  COS  a  —  h  _'p  sin  a  —  ;_^2_|_^_ 

-f 

_      1 

(27) 

9          ~          /          ^          c 

Whence    A;  =  ^  cos  a ^— ,     Z  =  ^  sin  a 

(X  — p  0 

— 

/ 
a  +6' 

(28) 

and                                 W^V-f^      \_ 

(29) 

Then  from  (28)  and  (29)  we  get       • 


(30) 


^       2(a  + 6)(^  COS  a  4-/sin  a) 
^'+/'  — c(a  +  6) 

Finally,  substituting  (26)  and  (31)  in  (28)  gives 


2{a-\-h){gVh-^rSVa)    ' 

;  _  v'^[^^  +  /^-c(a  +  6)]--/ 
2(a  +  6)(^T/6+/l/a) 


(32) 


In  deriving  (30)  from  (28)  and  (29)  we  obtain  a  quadratic 
equation  in  which  the  coefficient  of  p^  becomes  zero,  and  there- 
fore one  root  is  infinite  (§  98,  III.).  Hence  one  directrix,  and 
consequently  one  focus  (28),  of  a  parabola  is  infinitely  distant 
from  the  origin. 

The  student  should  now  carefully  observe  the  correspondence  between 
the  results  here  found  algebraically  from  the  discussion  of  the  general 
equation,  and  those  obtained  geometrically  from  the  study  of  the  figure 
in  §  106. 

110.  To  determine  by  an  examination  of  the  coefficients  what  kind 
of  a  conic  is  represented  by  the  general  equation 

ax'-^  2hxy  +  bf-\-  2gx  +  2fy  +  c  =  0. 
From  equation  (12)  of  §  109  we  have 

e*  sin^  a  cos^  a  _  1  —  e^ 
h^  -  ab  —  h^' 


110.]  CONIC   SECTIONS.  166 

Since  the  first  member  of  this  equation  is  always  positive, 
^  —  1  and  l^  —  ah  must  always  have  the  same  sign,  and  must . 
vanish  together.     We  therefore  have  the  following  conditions: 

For  an  ellipse,     e  <  1,     e^  <  1.  .*.     /i'  <  ah. 

For  a  parabola,     e  =  1,     e^  =  l.  .-.    W  =  ah. 

For  a  hyperbola,     e  >  1,     e^  >  1.         .*.    h^  >  ah. 

When  e  =0,  equation  (10),  §  109,  gives 

1  _1_0 
a       h      W 

Therefore,  for  a  circle,     e  =  0,     a  =  6,     A  =  0.     (  Qf.  §  32. ) 

When     a +  6  =  0,     e  =  |/2.      [(15),  §  109.] 

The  conic  is  then  called  a  Rectangular  Hyperbola.* 

E.  g..,  a;^  —  y^—  a^  and  xy  =  K  are  rectangular  hyperbolas. 

When  c'=  0,  then  A  =  0  [(9),  §  109] ,  and  therefore  equations 
(5)  and  (1)  of  §  109  represent  two  straight  lines,  real  oi*  imagi- 
nary.    (See  also  §  54. ) 

Jf  A  =  0,  and  also  ah  —  h^  =  0,  then  &  is  not  necessarily  zero. 

The  first  three  terms  of  equation  (5),  §  109,  are  then  a  perfect 
square.     The  equation  may  therefore  be  written 

Vax  -j-  Vby  dz  \/^&=  0, 
and  represents  two  parallel  lines  which  coincide  when  c'=  0. 

For  convenience  these  results  are  collected  in  the  foUow^ing 
t«,ble : 

Curve.  Condition. 

Ellipse.  (e<l)     h^<ah. 

Parabola.  (e  =  l)     h^  =  ah. 

Hyperbola.  (e  >  1)     ^^  >  a6. 

Circle.  (^e  =  0)     a  =  h,  and  ^  =  0. 

Rectangular  Hyperbola.  (e  =  i/2)     a  -\-  h  =  0. 
Two    real    or    imaginary 

straight  lines.  /\=ahc^  2fgh  —  af — bg^ —  ch^  =0. 
Two  parallel  or  coincident 

straight  lines.  A  =  0,     and    h^=^  ah. 

♦See  §§116  and  121. 


166  CONIC   SECTIONS.  [110. 

EXAMPLES. 

1.  Show  that  the  directrices  and  the  foci  of  a  parabola  are  infinitely 
distant  from  the  centre. 

2.  Show  that  the  foci  of  a  circle  coincide  with  the  centre,  while  its  di- 
rectrices are  infinitely  distant  from  the  centre. 

3.  Show  that  when  the  conic  is  two  distinct  intersecting  lines,  the  foci 
coincide  with  their  point  of  intersection;  and  that  the  directrices  both 
pass  through  this  point. 

4.  When  the  conic  is  two  parallel  lines  (limiting'  case  of  parabola), 
show  that  the  centre,  the  foci,  and  the  position  of  the  directrices  are  inde- 
terminate ;  but  the  directrices  are  perpendicular  to  the  two  lines. 

5.  If  the  general  equation  represents  a  parabola,  show  that  the  centre 
is  at  infinity. 

6.  If  a  =  b,  show  that  the  principal  axis  is  equally  inclined  to  the  axes 
of  coordinates. 

7.  If  a  =  &  and  /i  =  0,  show  that  the  direction  of  the  principal  axis  is 
indeterminate. 

8.  lth  =  0,  show  that  the  principal  axis  of  the  conic  is  parallel  or  per- 
pendicular to  the  X-axis. 

Find  the  equations  and  trace  the  conies,  having  given 

Directrix. 
9.    2x-\-y  =  2. 

10.  3x  — 2/=3. 

11.  x  —  y  =  2. 

12.  2x~y  =  l. 

13.  3x-\-iy-\-10  =  0. 

14.  3x4-2/ =  5. 

Find  the  parameters  and  trace  the  following  conies : 

15.  x^  — 16x2/— 112/2 +  2Cte  + 502/  — 35  =  0.       Ans.    e  =  2,  «  =  Jtan-^. 

16.  14x^  +  2x2/  +  1%'  —  32x  +  S2y  +  29  =  0. 

17.  3x2  — 8x2/  — 32/2— 2OX  +  IO2/  — 5  =  0. 

What  do  the  following  equations  represent? 

18.    y''  +  ax  —  2ay  =  0.  19.    xy -\- ax  —  ay  =  0. 

20.    x2-f-2x  — 42/-|-l  =  0.  21.     (x  — 3/)»=a(a;4-y). 


Focus. 

6. 

(2,2) 

1. 

(2,0) 

3 

(3,1) 

2. 

(0,0) 

VS. 

(-1,1) 

h 

(2,-1) 

1. 

111.]  COXIC  SECTIONS.  167 

22.     {x-\-yy-\-(x  —  yy=2a\         23.    y^  —  x''-{-2ax  =  0. 

24.    3x2  +  4x2/  +  2y'  =  2.  25.    42?^  —  Sxy  —  ^y^  =  l. 

26.    Show  that  if  the  origin  is  at  the  centre  and  the  principal  axis  is 
taken  as  the  x-airis,  the  equation  of  a  conic  may  be  written 

ax^-\-by^  =  i. 

Tangents. 

111.     To  find  the  equation  of  the  tangent  to  the  conic  represented  by 
the  general  equation 

ax'  +  2hxy  -f  bf  +  2gx-^2fy  +  c  =  0.  (1) 

The  equation  of  the  tangent  to  any  curve  f{x,  y)  =0  a>t  the 
point  (a;',  y')  is  (§  85) 

y-y'=%i^-^l'  (2) 


For  equation  (1)  we  have  found  in  §  84 

dy  ^       ax  -{-  hy  -^g 
dx  hx  ^  by  ^  f 

Therefore  the  required  equation  is 


(3) 


y  y-   hx'-i-by'^s^''  "'^^  ^^^ 

or         axx'-^  h(xy'-{-  x'y)  +  byy'-^  gx  -\-fy 

=  ax''^  2hxY-{-  by''-}-  gx'+fy'.      (5) 

Add  gx'-\-fy'-{-  c  to  both  sides  of  (5)  ;  then,  since  (x',  y')  is  on 
the  conic,  the  right  member  will  vanish  and  we  have  the  required 
equation, 

axx'-^  h{xy'^  x'y)  +  byy' -\-  g(x  +  x')  ^f(y  -\-y')-^c  =  0.      (6) 

Observe  that  the  equation  of  the  tangent  at  (a;',  y')  is  obtained 
from  the  equation  of  the  conic  by  writing  xx^  for  x^,  x'y  +  xy'  for 
2xy,  yy^  for  ^/^,  x  -\-  a/  for  2x,  and  y  -{-y'  for  2y.  Note  also  that 
putting  x  for  re'  and  y  for  y'  in  (6)  reproduces  the  equation  of  the 
curve. 

E.  gr.,  the  equation  of  the  tangent  to  the  parabola  y'^=  4ax  at  (x^,  y^)  is 
2/3/'=2a(x  +  xO. 


168  CONIC   SECTIONS.  [112. 

112.  Two  tangents  can  he  drawn  to  a  conic  from  any  point,  which 
will  he  real,  coincident,  or  imaginary,  according  as  the  point  is  outside, 
on,  or  within  the  curve. 

Let  the  equation  of  the  conic  be  [§  108,  (6)] 

ax'-^f-^2gx^-g'=0,  (1) 

where  a  =  1  —  e^ 

Let  {h,'k)  be  any  point;  then  the  equation  of  any  line  through 
this  point  will  be  (§  46) 

y  —  k  =  m(x — h).  (2) 

Eliminating  y  between  (1)  and  (2)  gives 
(a  +  m'')x'^2(km  —  hm'-\-  g)x  +  hW  —  2hhm  -^Tc" -{- g""  =  0.    (3) 

The  roots  of  (3)  are,  by  §  24,  the  abscissas  of  the  points  of  inter- 
section of  (1)  and  (2).  If  these  roots  are  equal,  the  points  of 
intersection  will  coincide  and,  by  §  78,  (2)  will  be  tangent  to  (1). 
The  condition  that  (3)  shall  have  equal  roots*  is 

{km  —  hm^^gy  =  {a^m^){hV  —  2hhm^h''-^g''),       (4) 
or    {ah^'-Y  2gh  -f  g'')m'~2{ahk  +  gk)m  +(aF+  a^'  —  g')=0.  (5) 

Equation  (5)  is  a  quadratic  in  m  whose  roots  are  the  slopes  of 
the  tangents  from  {h,  k)  to  the  conic.  Since  a  quadratic  equation 
has  two  roots,  two  tangents  will  pass  through  any  point  Qi,  k). 

The  conic  is,  therefore,  a  curve  of  the  second  class. 

The  roots  of  (5)  are  real,  equal,  or  imaginar}^,  according  as 

ah'-i-  F+  2gh  +  g'  >,  =,    or  <  0.  (6) 

Therefore  the  tangents  are  real,  coincident,  or  imaginary  ac- 
cording as  the  point  (A,  k)  is  outside,  on,  or  within  the  conic. 
(§20,11.) 

Since  equation  (3)  is  a  quadratic  in  x,  any  straight  line  meets 
a  conic  in  two  points,  which  may  be  real,  coincident,  or  imaginary. 

Therefore  the  conic  is  also  a  curve  of  the  second  order. 

If  e  =  1  and  m  =  0,  then  a  -f-  m^=  0,  and  hence  one  root  of  (3) 
is  infinite  (§  98,  III.).  Therefore  a  straight  line  parallel  to  the 
axis  of  the  parabola  meets  the  curve  in  one  point  at  a  finite  dis- 
tance, and  in  another  at  an  infinite  distance  from  the  directrix. 

*The  two  roots  of  ax^  +  6a;  +  c  =  0  will  be  equal,  if  6^  =  4ac 

The  method  here  used  is  worthy  of  special  attention  because  of  its  wide  application.  To 
And  the  condition  that  any  two  curves  shall  touch  we  may  treat  their  equations  simultane- 
ously and  eliminate  one  variable,  and  then  take  the  condition  for  equal  roots. 


113.]  conic  sections.  169 

Pole  and  Polar. 
113.     The  equation  of  the  tangent  to  the  conic 

ax'^  2hxy  +  bf-\-  2gx  +  2/2^  -f  c  =  0  (1) 

at  the  point  (x',  y')  is  (§  111) 

axx'^Kxy'+x'y)^hyy'^g{x^-x')^f{y^y')-^c  =  0.   (2) 

Suppose,  however,  that  P'{x^,  y')  is  not  on  the  conic.  This 
equation  still  has  a  meaning,  still  represents  a  straight  line  related 
in  a  definite  way  to  the  point  (x',  y')  and  the  conic  (1).  This 
line  will  cut  the  conic  in  two  points  (§  112). 


Let  these  points  be  Pi(iCi,  2/1)  and  PaCrca,  y^)' 

Then  the  equations  of  the  tangents  at  these  points  are  (§  111) 

axx^^  h(xyi-{-  x,y)-{-hyy,-{- gi^x  -f  x,)-\-f(^y  +  1/1)+  c  =  0,    (3) 

axx.,-\-  h(xy,-\-  x,y)  +  byy^-^  g(x  +  x,) -\-f(y  +  2/2)  +  c  =  0.    (4) 

The  conditions  that  (3)  and  (4)  shall  pass  through  {x',  y')  are 

ax'x,-^  Kx%-^  a^y  ) -f  hy'y,^  g{x' -\-  x,)-^f{y'-^  2/i)  +  «  =  0,     (5) 

ax'x^-\-  h{x'y2-{-  X2y')-{-  by'y,-\-  g(x'+  a?2)  +  /(2/'+  2/2)+  ^  ==  0.     (6) 

But  (5)  and  (6)  are  also  the  conditions  that  (2)  shall  pass 
through  both  of  the  points  (^Xi,  i/i)  and  (a?2,  2/2)- 

Therefore  (2)  is  the  line  passing  through  the  points  of  contact 
of  the  tangents  from  the  point  P'(x',  y'). 

The  point  {x' ,  y')  and  the  line  (2)  are  called  Pole  and  Polar 
with  respect  to  the  conic  (1). 


170  CONIC   SECTIONS.  [114. 

The  tangents  from  the  point  {x\  y')  will  be  real  or  imaginary 
according  as  {x'^  y')  is  outside  or  inside  the  conic  (§  112);  but 
the  line  (2)  is  real  when  (a?',  y')  is  real.  So  that  there  is  always 
a  real  line  passing  through  the  imaginary  points  of  contact  of  the 
two  imaginary  tangents  drawn  from  a  point  within  a  conic. 

If  (x',  y')  is  on  the  conic,  the  two  tangents  from  it  will  coin- 
cide, and  each  of  the  points  (^j,  y^)  and  (X2,  i/2)  will  coincide 
with  (x',  y').  Therefore  the  tangent  is  the  particular  case  of  the  polar 
which  passes  through  its  own  pole. 

1 14.  If  the  polar  of  a  point  P' {x' ,  y')  pass  through  F"(x",  y"), 
then  will  the  polar  of  P"  pass  through  P' .      (See  ^^,  §  113.) 

Let  the  equation  of  the  conic  be  [§  112,  (1)] 

ax?^-  2/^+  2gx  +  /=  0.  (1) 

The  equations  of  the  polars  of  P'  and  P"  are 

axx'+yy'^-g{x-^x')-^g'=0  (2). 

and  axx"^yy"^g{x-^x")^g''=0.    (§113.)     (3) 

The  line  (2)  will  pass  through  the  point  P"  if 

ax'x" -^  y'y"-^  g(ix'^  x")-\- g''=0]  (4) 

but  this  is  also  the  condition  that  (3)  shall  pass  through  P',  which 
proves  the  proposition. 


Cor.  I.      The  loeus  of  the  poles  of  all  lines  parsing  through  a  fixed 
point  is  a  straight  line;  viz.,  the  polar  of  the  fixed  point. 

Cor.  II.     If  the  polars  of  two  points  P  and  Q  meet  in  B,  then  B  is 
the  pole  of  the  line  PQ. 


115.] 


CONIC  SECTIONS. 


171 


Two  straight  lines  are  said  to  be  conjugate  with  respect  to  a 
conic  when  each  passes  through  the  pole  of  the  other. 

Two  points  are  said  to  be  conjugate  with  respect  to  a  conic  when 
each  lies  on  the  polar  of  the  other. 


115.*     Any  chord  of  a,conic  passing  through  a  point  0  is  divided 
harmonically  by  the  conic  and  the  polar  of  0. 


Let  OQ  be  any  line  cutting  the  conic  in  P  and  Q  and  the  polar 
of  0  in  E.  We  are  to  prove  that  the  line  PQ  is  divided  har- 
monically ;  i.  e.  that  it  is  divided  internally  and  externally  in  the 
same  ratio  at  B  and  0.     We  must  therefore  prove 

PR 


whence 


OP 
PR 


OP 

0Q~  RQ' 

OQ        OP^OQ        OP^OQ 


RQ       PR-\^RQ        OQ—OP 
From  the  first  and  last  ratios  by  composition, 
OP^PR  20q 


OP 
OP^OQ 


OP^OQ 
2 


OP'OQ         OP-\-PR' 


JL  .  J___l_* 
OP'^  OQ 


OR' 


(1) 


*0B  is  called  the  Harmonic  Mean  between  OP  and  OQ. 


172  CONIC  SECTIONS.  [116., 

Take  0  for  origin  and  tlie  line  OPQ  for  the  ic-axis ;  let  the 
equation  of  the  conic  be 

ax'^  2hxy  +  hif-^  2gx  -\-2fy-^c  =  0.  (2) 

Then  the  equation  of  AB,  the  polar  of  0,  is  (§  113) 

9x-{-fy^G  =  0.  (3) 

Where  the  line  ^  =  0  cuts  AB  we  have 

.  =  _l=Oie,  i.e.^  =  -f.  (4) 

Where  the  line  y  =  0  cuts  the  conic  we  have 

aa:^+23a;  +  e  =  0,orl  +  2|i+|  =  0.  (5) 

(§91)     (6) 
(V) 


•  •      OF    ^    OQ 

_       ^9 
c 

From  (4)  and  (6)  we  get 

1      ,      1 

2 

OP    ^   OQ 

~  OB' 

which  is  the  same  as  (1). 

Asymptotes,  Similar  and  Conjugate  Conics. 

116.*    Consider  the  three  concentric  conics  whose  equations 
are 

ax'+  2hxy  -{-hy^=c,  ( 1 ) 

aa?2-f  2hxy  +  6i/'  =0,  (2  ) 

and                               ax^^2hxy -{-hy"^^  —  c;  (3) 

and  let  h?  >  a6,  so  that  all  the  loci  are  real. 

Solving  these  equations  for  y  gives,  respectively, 

y  =  —  ^x±  ^\/\h'—ab)x'^bc  =  —  ^x±  Aj,  (4) 


y  =  —  ^x±^V{h'  —  ab)x'-\-  0  =  —  ^x±X,  (5) 

r  -|  7 

'^x  ±  -\/{h'  —  ah)x'—he^~^ 


h         1    / h 

and  y  =  —  -x  ±  j-V{h''  —  ab)x''—bc  =  --j-x±^,,  (6) 


116.]  CONIC   SECTIONS.  173 

where  /j,  ^,  and  I2  ^^^  P^t  for  the  terms  containing  the  radicals 
and  are  quantities  such  that  for  finite  values  of  ic,  A^  >  A  >  Ag ;  but 
for  infinite  values  of  ic,  Xy=X=  x^.  Therefore,  in  the  finite  part 
of  the  plane,  the  loci  represented  by  equations  (1)  and  (3)  lie  on 
opposite  sides  of  the  two  lines  given  by  equation  (2 ) ;  but,  at  an 
infinite  distance  from  the  origin  (the  centre  of  the  conies),  the 
ordinates  of  the  three  loci  are  equal  and  the  three  loci  come 
together. 


The  values  of  y  are  real  for  all  values  of  x  m  equations  (4)  and 
(5),  but  in  (6)  y  is  imaginary  when  x'^(h'^  —  «&)<  he.  The  three 
conies  are  as  shown  in  the  figure,  where 

D'A=AD  =  X„     CA=AC  =  X,     B'A  =  AB  =  X,; 

and  OA  is  the  line 

y  =  -^oc.  (7) 

Thus  OA  bisects  all  chords,  BB',  CC,  DD',  parallel  to  the  y-Sbxis ; 
i.  e.  the  line  (7)  is  a  diameter  of  each  of  the  three  given  conies. 
(See  §126.) 


174  CONIC   SECTIONS.  [116, 

The  straight  lines  00  and  OC  which  meet  the  conies  at  inj&nity 
are  called  Asymptotes  * 

Therefore  equation  (2)  represents  the  asymptotes  of  the  conies 
given  by  both  (1)  and  (3),  and  we  see  that  the  asymptotes  of  a 
conic  pass  through  its  centre. 

When  we  say  above  that  the  ordinates  of  the  three  loci  become 
equal  when  a?  =  oo  ,  we  mean  that  their  difference  bears  a  vanish- 
ing ratio  at  last  to  any  namable  finite  quantity.  Two  parallel 
lines  are  said  to  come  together  in  the  sense  that  the  distance  be- 
tween them  at  last  bears  a  vanishing  ratio  to  the  distance  gone 
along  them.  In  this  sense  any  parallel  to  an  asymptote  meets  the 
hyperbola  where  the  asymptote  does.  This  parallel,  however, 
meets  the  hyperbola  elsewhere  in  the  finite  region.  Now  suppose 
such  a  parallel,  keeping  its  slope,  to  move  up  to  coincidence  with 
the  asymptote,  the  finite  intersection  moves  along  the  curve  and 
goes  out  to  infinity.  Thus  the  asymptote  meets  the  hyperbola  in 
two  points  at  infinity.     (See  §  146.) 

Let  20  be  the  angle  between  the  lines  represented  by  equation 
(2),  then  (§  57)  

tan2<?  = (8) 

a  +  6  ^     , 

If  h?  <  ah,  these  lines  are  imaginary,  and  the  loci  of  (1)  and 
(3)  are  ellipses  (§  110).  Therefore  the  ellipse  has  no  real 
asymptote. 

If  h?  =  ab,  equation  (1)  represents  two  parallel  lines  equidis- 
tant from  the  origin,  (2)  represents  two  coincident  lines  midway 
between  them,  and  (3)  represents  two  imaginary  lines. 

If  a  -)-  6  =:  0,  the  asymptotes  are  perpendicular  to  each  other. 

A  conic  whose  asymptotes  are  at  right  angles  is  called  a 
Rectangular  Hyperbola  (§§  110, 121). 

Similar  Conies.  Two  curves  are  similar  when  the  one  is  merely  a 
magnification  of  the  other;  i.  e.  when  we  could  get  the  equation ,  of 
the  one  from  that  of  the  other  by  merely  changing  rectangular 
axes  and  scale. 

E.g.^  the  equation  of  any  circle  can  be  reduced  to  x^-\-y'^=i  by  movlng^ 
the  origin  to  its  centre  and  taking  its  radius  for  the  unit  of  the  scale. 

♦Greek,  asymptotos,  not  falling  together. 


116.]  CONIC  SECTIONS.  175 

Let  k  be  the  factor  of  magnification ;  then  for  two  similar  conies, 
we  have 

P'r=k'PF    and     P'S'=k' PS. 

PF'      PF 
•'■    P^'=PS='-  [(2),  §108.] 

That  is,  similar  conies  have  the  same  eccentricity;  and,  con- 
versely, conies  having  the  same  eccentricity  are  similar. 
Hence  all  parabolas  are  similar. 
From  equation  (8)  we  have  by  Trigonometry 


tan  2.  =  /*^°t.  =2±:?55!^  =  ^^^^—b         (9) 
1  — tan^^  2  — sec' 6/  a -\- b 

/2  — sec'^V_l  — sec'g 
•*•     \     a+b     )~    ab  —  h'  '  ^^"^ 


Solving  (10)  gives  [c/.  (13),  (14),  and  (15),  §  109] 
1  1  a  +  6  1 


(11) 


sec'o       2      2i/(a_6)2^4A2-e2- 

.-.     sec^  =  e.  (12) 

That  is,  the  eccentricity  of  a  hyperbola  is  equal  to  the  secant  of  half 
the  angle  between  its  asymptotes.  Hence  all  hyperbolas  having  the 
same  asymptotes,  and  lying  within  the  same  angle,  have  the  same 
eccentricity,  and  are  therefore  similar,  i.  e,  similar  to  the  asymp- 
tote-pair. 

Conjugate  Hyperbolas.  If  A^>  a6,  both  roots  of  (11)  are  posi- 
tive. Since  (11)  involves  only  a,  6,  and  h,  and  is  therefore  the 
same  for  equations  (1),  (2),  and  (3),  these  two  positive  roots 
give  the  eccentricities  of  the  two  hyperbolas  (1)  and  (3);  and 
also  the  secants  of  half  the  supplementary  angles  between  their 
common  asymptotes  (2). 

For  the  same  reason,  the  directions  of  the  principal  axes  (§  109) 

of  the  two  hyperbolas  are  determined  by  the  values  of  a  which 

satisfy  the  equation  [(17),  §  109] 

2A  ,^„^ 

tan  2a  = -.  (13) 

a  —  0  ^ 

But  these  values  of  a  differ  by  90° ;  therefore  the  principal  axes 
of  the  two  hyperbolas  are  perpendicular  to  each  other. 


176  CONIC   SECTIONS.  [117. 

Equation  (13)  with  (5)  of  §  58  show  that  the  axes  are  the  bi- 
sectors of  the  angles  between  the  asymptotes. 

From  equation  (22),  §  109,  we  see  that  the  foci  of  the  two 
hyperbolas  are  equidistant  from  their  common  centre. 

Two  conies  having  these  relations  will  be  found  to  satisfy  the 
definition  of  Conjugate  Hyperbolas  given  in  §  140. 

If  h^<^ab,  equation  (11)  has  only  one  positive  root.  Hence 
an  ellipse  has  no  real  conjugate. 

If  c  be  arbitrary,  equations  (1)  and  (3)  each  represent  a  system 
of  similar  hyperbolas,  such  that  for  each  conic  in  one  system  there 
is  a  corresponding  conjugate  in  the  other  system.  Moreover,  the 
asymptotes  are  the  limiting  forms  of  both  systems*  corresponding 
to  c  =  0 ;  i.  e.  two  intersecting  lines  are  a  pair  of  self-conjugate 
hyperbolas.     (See  §  147.) 

Ex.  Show  that  the  sum  of  the  squares  of  the  reciprocals  of  the  eccen- 
tricities of  two  conjugate  hyperbolas  is  equal  to  unity. 

1 17.*  To  find  the  equation  of  the  asymptotes  of  a  conic;  also  the 
equation  of  its  conjugate. 

Let  the  equation  of  the  given  conic  be 

ax'-i-  2hxy  +  hif-\-  2gx  +  2///  +  c  =  0,  (1) 

Write  down  the  two  equations 

ax'^-  2hxy  +  bif  +  2gx  +  2/i/  +  c  -  -^^,  =  0,  (2) 

and         ax'-\-  2hxy  +  %^+  2gx  -f  2/?/  +  c  -  -^^,  =  0.  (3) 

These  three  conies  are  concentric  [(7),  §  109].  Moving  the 
origin  to  the  centre  without  changing  the  direction  of  the  axes, 
we  get  [(5)  and  (9),  §  109],  respectively, 

ax'^  2hxy  +  %^+  ^^^,  =  0,  (4) 

ax'^2hxy-\-by''  =  0,  (5) 

and  ax'-\-  2hxy  +  by'  —  ^^  ^  ^,  =  0.  (6) 

*  For  the  different  manner  in  which  the  asymptotes  are  described  when  considered  as 
conies  belonging  to  the  two  different  systems  see  §  159,  III. 


117.]  CONIC  SECTIONS.  177 

Now,  if  ^^>  a6,  (4)  and  (6)  are  conjugate  hyperbolas  (§  116), 
while  (5)  represents  their  common  asymptotes.  Therefore  (2) 
and  (3)  are  the  required  equations. 

Cob.     The  lines  represented  by  the  equation 

aoif-^2hxy  +  bu'=0 

are  parallel  to  the  asymptotes  of  the  conic  represented  by  the  general 
equaiion  (1). 

EXAMPLES. 
Find  the  equations  of  the  tangent  and  normal  to 

1.    a;2  =  23/,  at(— 2,  2).  2.    j/^^&c,  at  (2,  —  4). 

3.    a;2_|_2,2^25,  at(4,  —  3).  4.    x^  — i/2=i6,  at  (— 5,  3). 

5.    x»  +  42/2=8,  at(— 4,  3).  6.    2y2_a;^=  4,  at  (2,  —  2). 

7.  2/^  +  4x  +  2^  +  l  =  0,at(-4,3). 

8.  3x2  _|.  52.y  _  22,2^  0,  at  (1,  3)  and  (—  2,  1). 

9.  2x2  — 4x3/  +  2/'  +  2x  — 4i/  — 15  =  0,  at(2,  3). 

10.    x2  +  3x2/  +  4i/2_3x  +  52/  +  9  =  0,  at(4,  —  1). 

Find  the  equations  of  the  tangents  to  each  of  the  following  conies  at  the 
origin: 

11.    2x2  + 33/2+ 2x  =  0.  12.    x2  +  2x  +  3i/  =  0. 

13.    2x2/ +  5x  —  31/ =  0.  14.    3x2  — 2x2/+4x-22/ =  0. 

15.    x2  +  2x2/  — 31/2  +  42/ =  0.  16.    ax2  +  2/ix2/ +  62/' +  2c/x  +  2/2/ =  0. 

17.    State  a  rule  for  finding  the  tangent  to  a  conic  at  the  origin. 

Find  the  polar  of  the  point 

18.  (3,  2)  with  respect  to  y^=  6x. 

19.  (—  2,  —  4)  with  respect  to  x^  +  2/2=  4. 

20.  (1,1)  with  respect  to  2x»  +  32/2=  1. 

21 .  (0,  0)  with  respect  to  2x2  _  3^,2  _|_  43.  _  3^,  _j_  4  _  q. 

22.  (—1,2)  with  respect  to  3x2  —  6x2/  +  2/'  —  2x  +  42/  +  3  =  0.* 

23.  (3,-1)  with  respect  tox'  +  2x2/  +  32/''  +  4x  —  6y  + 1  =  0. 

24.    Give  a  general  rule  for  writing  the  equation  of  the  polar  of  the 
origin. 
13 


178  CONIC   SECTIONS.  [117. 

25.  Find  the  tangent  to  the  parabola  y'^=  6x  which  makes  an  angle  of 
45°  with  the  x-axis. 

26.  Find  the  equations  of  the  tangents  to  the  parabola 
y^  —  Ax-\-2y-\-\  =  0  whose  slopes  are  2,  and  J. 

27.  Find  the  tangents  to  the  conic  ia;^  +  2/^=  ^  whose  slope  is  J. 

28.  Find  the  equations  of  the  lines  which  have  the  slope  ( — J),  and 
touch  the  conic  2x^  +  2/^  —  ^2/  + 1  =  0. 

29.  Find  the  equation  of  the  normal  to  the  conic  {y  —  Sy  +  4(x  -(- 1)  =  0> 
parallel  to  the  line  2y  -\-Sx  =  0. 

30.  Find  the  normals  to  the  conic 

9(22/  +  ly  —  4(3a;  —  2)«=  36  whose  slope  is  3. 

Find  the  tangents  to  the  following  conies  drawn  from  the  given  points 
(see  §112): 

31.  y'=ix,    (2,3).  32.    y'=6x,     (-3,-1). 

33.  a;2  +  2/2=25,    (—1,7).  34.    9x^+252/2=225,    (10,-3). 

35.  2/'  +  8a;-42/  +  4  =  0,    (-2,2).      36.    x'-{-xy  +  y'=i2,    (-3,4). 

37.  (x  + 2)2 +  2(^-2)2=  27,    (1,-1). 

38.  Show  that  the  polar  of  the  focus  is  the  directrix. 

What  is  the  locus  of  the  intersection  of  tangents  at  the  ends  of  focal 
chords?    (Use  equation  (6),  §  108.) 

39.  Show  that  the  line  joining  the  focus  to  any  point  on  the  directrix  is 
perpendicular  to  the  polar  of  the  latter  point. 

40.  Show  that  tangents  to  a  conic  at  the  ends  of  a  chord  through  the 
centre  are  parallel.     (See  Ex.  26,  p.  167.) 

41.  "What  is  the  polar  of  the  centre  of  a  conic?  Where  is  the  pole  of 
a  line  passing  through  the  centre  ? 

42.  What  is  the  pole  of  x  cos  a-\-y  sma=p  with  respect  to 

a.2_|_2,2^y29     y'i  =  2x? 

43.  Show  that  if  the  slope  of  a  variable  chord  of  a  conic  is  constant  the 
tangents  at  its  extremities  always  intersect  on  the  same  diameter.  State 
the  converse.  From  the  definition  of  conjugate  lines  in  §  114,  what  may  we 
call  this  diameter  and  the  diameter  parallel  to  the  variable  chord  ? 

44. .  If  mi  and  rrh  are  the  slopes  of  the  two  diameters  mentioned  in  43, 
and  ax"^  -\-by^=  1  is  the  equation  of  the  conic,  show  that  miw^  =  —  j-, 

45.  If  /S  =  0  and  5i^=  0  are  the  equations  of  two  conies,  what  is  the  locus 
represented  by  the  equation 


118.]  CONIC   SECTIONS.  179 

46.  Find  the  equation  of  the  conic  passing  through  the  point  (0,  2)  and 
the  common  points  of 

y^  =  ix  and  (x  —  4)»  =  2(^  +  5). 

What  kind  of  a  conic  is  it  ? 

47.  Find  the  equation  of  the  conic  passing  through  the  origin  and  the 
common  points  of 

a;2_3a;2/  +  2/'4-10a;  — 102/4-21=0  and  4a;2  +  9?/^  +  16x  —  362/ + 16  =  0. 

48.  If  two  conies  have  their  principal  axes  parallel  their  points  of  inter- 
section lie  on  a  circle. 

49.  Find  the  equation  of  the  circle  passing  through  the  common  points 
of 

9a;2^4y2_|_i8x  — 242/  +  9  =  0    and    x''  —  y^-{-2x-{-iy  —  4:  =  0. 


Standard  Equations  of  the  Conic  Sections. 

118.  Let  the  directrix  be  the  i/-axis,  the  principal  axis  of  the 
conic  (§  109)  the  ic-axis,  and  let  k  denote  the  distance  from  the 
directrix  to  the  corresponding  focus.  Then  the  equation  of  any 
conic  takes  the  simple  form  [(6),  §  108] 


or  (1  —  e')x'  -hy'  —  ^kx  -^k'  = 


0.} 


(1) 


li  x  =  Om  (1),  then  y  =  ±  kV — 1. 

Hence  a  conic  does  not  intersect  its  directrix. 

If  2/  ^  0,  then  there  are  two  real  values  of  x,  viz. , 

_     k  _     k 

Therefore  a  conic  section  cuts  its  principal  axis  in  two  points. 
These  points  are  called  the  Vertices  of  the  conic.  The  centre 
is  midway  between  the  vertices. 

The  Latus  Rectum  of  a  conic  is  the  chord  through  either 
focus  perpendicular  to  the  principal  axis. 

To  find  its  length,  let  a?  =  A;  in  (1),  then 

y  =  ±:  ekj     and    2y  =  Latus  Rectum  =  2ek. 

The  different  cases  corresponding  to  the  different  values  of  e 
will  now  be  separately  considered. 


180 


CONIC   SECTIONS. 


[119. 


119.    The  Parabola.     e  =  l. 

When  6  =  1,  equations  (2)  of  §  118  give 


x,=  ik=DO, 


OCq 


0 


00 


Hence  the  parabola  has  one  vertex  midway  between  the  focus 
and  directrix,  and  the  other  at  infinity.* 


When  e  =  l,  equation  (l)of§  118  gives  for  the  equation  of  the 
parabola  referred  to  its  axis  and  directrix 

f=2k(ix-ik).  (1) 

Let  a  =  ik  =  DO=  OF;  then  this  equation  becomes 

i/'  =  4a(aj  — a).  (2) 

Now  write  x  -\-ain  the  place  of  x ;  this  moves  the  origin  to  the 
vertex  0(a,  0)  [§  66,  (10)],  and  the  equation  becomes 

f  =  4:ax,  (3) 

which  is  the  standard  form  of  the  equation  of  the  parabola. 
When  x  =  am  (3),  y=zt2a. 

.  • .     L'L  =  4:a  =  Latus  Rectum. 
Ex.  1.    Trace  the  parabolas  y^  =  —  iax  and  a;*=  rb  ^ay. 
Ex.  2.    Construct  the  parabola,  having  given  the  focus  and  the  directrix. 
♦Compare  this  result  with  the  position  of  B  in  the  figure  of  §  106  when  a  =  /?. 


120.]  CONIC   SECTIONS. 

120.     The  Ellipse.     e<l. 


181 


When  e  <  1,  the  two  ic-intercepts  [(2),  §  118]  are  both  finite 
and  positive ;  that  is, 


1+6 
k 


oc^- 


DA'  >  k. 


Hence  the  ellipse  has  two  vertices  lying  on  the  same  side  of  the 
directrix,  but  on  opposite  sides  of  the  focus. 


c 

R 

B 

Y 

^ 

P 

R' 

X 

(] 

L 

P 

^ 

D 

■vj 

P 

O              Q    F'          lA' 
B' 

D' 

Let  0  be  the  centre,  and  let  AA'^=^  2a. 

k  k  2ek 


Then 


2a  =  a^a  —  Xi  = 


1  —  e        1+e       1  —  e' 


whence 


a  k 

k  = ae. 

e 


Also  D0  =  K«.+  a-,)=4(j^  +  j^) 


(1) 


(2> 


e 


(3) 
(4) 


182  CONIC  SECTIONS.  [120. 

Substituting  in  equation  (1)  of  §  118  the  value  of  h  given  by 
(2)  gives  for  the  equation  of  the  ellipse  referred  to  DC  and  DX 

(^-7  +  «ey+2/'  =  eV.  (5) 

The  origin  may  be  transferred  to  the  centre,  0(-,  O),  by  writ- 
ing a:;  +  -  in  the  place  of  x  [§  66,  (10)]  ;  this  gives 

6 

or  x\l  —  e')+f  =  a\l  —  e'). 

1.  (6) 


=  0 

in 

(6): 

x' 
a' 

,  we 

y 

1       '^^ 

When  X 

'   a^(l-0 
have 

=  ±aVl  — 

e'\ 

which  gives  the  ^-intercepts  OB  and  OB'. 
If  these  lengths  are  denoted  by  zt  6,  we  have 

b'  =  a\l-e^),  (7) 

and  equation  (6)  takes  the  standard  form 

—  4--^  =  i*  '  rs^ 

^2  i-  j2        A.  (^»; 

Since  e  <  1,  6  <  a  from  (7);  therefore 
B'B  <  AA', 

Hence  the  line  A  A'  is  called  the  Major  Axis,  and  BB'  is  called 
the  Minor  Axis  of  the  ellipse. 

Take  OF'^FO  and  OD'=^DO;  draw  D'C  perpendicular  to 
OX.  Then  F'  is  the  other  focus,  and  D'C  the  corresponding 
directrix  (§  109).  Hence  the  foci  are  the  points  F'(ae,  0)  and 
and  F( — ae,  0)  from  (4);  and  the  equations  of  the  directrices 
are,  from  (3), 

x  =  ±^.  (9) 

Let  P(Xy  y)  be  any  point  on  the  ellipse;  draw  a  line  through 
P  parallel  to  A  A'  meeting  the  directrices  in  B  and  B',  and  draw 
PQ  perpendicular  to  AA\ 

*  For  a  discussion  of  this  equation  see  §  35. 


120.]  CONIC  SECTIONS.  183 

Then  FP=e'EF, 

and  rP  =  e '  RP,  [(2),  §  108] 

/.    FP=e* DQ  =  e{DO  ^OQ) 

=  e\^'\-xj=a-\-ex,  (10) 

and  F'P  =  e'  Qiy=  e{  OD'—  OQ) 

=  6( x\=a  —  ex.  (11) 

Whence  FP  +  F'P  =  2a.  (  C/.  §  34. )     ( 12 ) 

From  equations  (7)  and  (4)  we  get 


ae 


y^a?—h'  =  FO=OF'. 


••     ' -a AA''  ^^^^ 

To  find  the  length  of  the  latus  rectum  we  put  £c  =  ±  ae  in  (8)  ; 
this  gives 

y'  =  h\\  —  e')=-,.  from  (7) 

a 

.-.     L'L  =  —,  (14) 

If  a  =  bj  equation  (8)  reduces  to 

x'  +  f  =  a\ 

and  equations  (13),  (4),  and  (3),  respectively,  give 

'   6  =  0,     FO  =  OF'=0,     DO  =  OD'=oo, 

That  is,  the  circle  is  the  limiting  form  of  the  ellipse  as  the 
eccentricity  approaches  zero,  and  the  directrices  recede  to  infinity. 

Ex.    Construct  an  ellipse,  having  given  the  foci  and  the  length  of  the 
major  axis. 

*  In  aU  conies  e  =  , .  ^'/^^"^^f  between  foci        ^^^   distances   become  Infinite   in  the 
distance  between  vertices 

parabola,  and  both  become  zero  in  the  case  of  two  intersecting  lines.    (See  also  (11),  §  121.) 


184 


CONIC   SECTIONS. 


[121. 


121.     The  Hyperbola,     e  >  1. 
From  equations  (2)  of  §  118  we  have  for  the  vertices 
h  ,  h 


X,  = 


and    Xo 


'       I  ^e  -'      1  —  e 

Since  e  >  1,  a?,  =  DA  <  k,  and  X2  =  DA'  is  negative. 
Therefore,  the  hyperbola  has  two  vertices  lying  on  the  same 
side  of  the  focus  but  on  opposite  sides  of  the  directrix. 


c 

Y 

C 

r, 

/ 

R' 

B                 ^^ 

R 

L 

A 

f 

X 

F^A' 

D' 

0                         D 
B' 

L' 

F               Q 

Let  0  be  the  centre,  and  let  A' A  =  2a. 


Then 


2a  =  A'D  +  DJL  = 

k       ,       k 


X2  ~r  ^1 
2ek 


e  —1    '   e  +  i       e'—  1 
and  k^ae 


•     e        e'—l 


D0  =  i(^,  +  X,)  =  i{Y^e  +  r^e) 


"l—e'  e 

FO  =  FD-\-DO=  —(k  +  j)= 


ae. 


(1) 

(2) 

(3) 
(4) 


121.]  CONIC  SECTIONS.  185 

The  equation  of  the  hyperbola  referred  to  DC  and  DX  is,  from 
(2),  and  (1)  of  §118, 

J^_ae  +  ^y+y^  =  «V.  (6) 

Moving  the  origin  to  the  centre  Oi ,  o)  gives 


Since  e  >  1,  the  quantity  a?{\  —  e")  is  negative;  iT  we  put 
—  b'  =  a\l  —  e'),ov 

h'  =  a\e'-l),  (7) 

equation  (6)  reduces  to  the  standard  form 

When  x  =  0,  i/  =  ±  b\/ —  1.  Since  these  values  of  y  are  both 
imaginary,  the  hyperbola  does  not  meet  the  line  through  its  centre 
perpendicular  to  its  principal  axis  in  real  points ;  but,  if  B,  B'  are 
points  on  this  line  such  that  B'O  =  OB  =  b,  the  line  BB'  is  called 
the  Conjugate  Axis.  The  line  AA'  joining  the  vertices  is  called 
the  Transverse  Axis. 

On  the  line  OX  take  OF'=FO,  and  OD'=DO]  then  F'  is  the 
other  focus  and  D'  C,  perpendicular  to  OX,  is  the  corresponding 
directrix  (§  109) .  Hence  the  coordinates  of  the  foci  are  (  zb  ae,  0) , 
from  (4),  and  the  equations  of  the  directrices  are,  from  (3), 

x=±-,  (9) 

6 

As  in  the  ellipse,  we  find  the  latus  rectum 

LL'=~,  (10) 

a  ■  ^ 

Equations  (7)  and  (4)  give 


(11) 


V'c 

=  Vd^ 

OF 

OF. 

I'^b' 

F'F 

a 

OA 

A'A 

•186  CONIC   SECTIONS.  [122. 

Let  P(x,  2/)  be  any  point  on  the  hyperbola ;  draw  a  line  through 
F  parallel  to  AA'  meeting  the  directrices  in  E  and  B',  and  draw 
PQ  perpendicular  to  AA'. 

Then  FP=e' PP,         F'P  =  e ' R'P.      [(2 ) ,  §  108] 

.-.     FP  =  e-D'Q=e(OQ—  OD)-^e(x—-\=ex  —  a;     (12) 

and   '  F'P  =  e'D'Q=e(iOQ-\-D'0)  =  e(x^^\  =  ex^a.    (13) 

Whence  F'P—  FP  =  2a.  ( Cf.  §  36. )     (14) 

Ji  a=:b,  the  equation  of  the  hyperbola  becomes 

x'  —  f  =  a\  (16) 

This  is  called  the  Equilateral  or  Rectangular  Hyperbola.  (See 
§§110,116.) 

Then  from  (11),  (3),  and    (4)  we  have,  respectively, 

e=|/2,      OZ)=4a|/2,      0F  =  ay'2. 

Ex.  Construct  a  hyperbola,  having  given  the  foci  and  the  distance  be- 
tween the  vertices. 

1 22.     Limiting  cases  of  conic  sections. 

lik  =  Oj  equation  (1)  of  §  118  reduces  to 

f=x\e'—l).  '  (1) 

This  equation  represents  two  straight  lines,  which  are  real  if 
e  >  1,  coincident  if  6  =  1,  and  imaginary,  but  with  a  real  point 
of  intersection,  if  e  <1. 

From  (2)  of  §  118  we  then  have  x-^,  =zx2=^0.  Hence  the  foci, 
the  vertices,  and  the  centre  of  two  intersecting  lines  all  coincide 
on  the  directrix.     The  two  directrices  also  coincide. 

When  e  =  cc  (a  being  finite),  the  equation  of  the  hyperbola 
[(8),  §  121]  reduces  to  x^  =  a^,  which  represents  two  parallel 
lines.  Equations  (3)  and  (4)  of  §121  then  show  that  the  foci  of 
two  parallel  lines  (considered  as  the  limiting  case  of  a  hyperbola) 
are  at  infinity  while  their  directrices  coincide  and  are  equidistant 
from  the  two  lines. 

Hence  we  must  consider  two  intersecting  lines,  real  or  imagi- 
nary (i.  e.  a  real  point),  two  coincident  lines,  and  two  parallel 
lines  as  limiting  cases  of  conic  sections.     (Qf.  %  107. ) 


CHAPTER  VIII. 

THE  PARABOLA. 

123.     Standard  equations  of  the  tangent,  polar,  ana  normal  to  the 
parabola. 

In  studying  the  properties  of  the  parabola  in  this  chapter  we 
shall  use  the  standard  form  of  the  equation  found  in  §  119,  viz., 

y^  =  4:ax.  (1) 

Then  the  focus  is  the  point  (a,  0),  the  directrix  is  the  line 
X  =  —  a,  and  the  latus  rectum  is.  4a. 

Equation  (6),  §  111,  applied  to  (1)  gives 

yy'=:2a(x-\-x^),  (2) 

as  the  equation  of  the  tangent  at  the  point  (x'j  y'),  if  (a/,  y')  is 
on  the  curve;  but  always  the  equation  of  the  polar  of  (a?',  y'), 
(§113),  with  respect  to  the  parabola  (1). 

The  equation  of  the  normal  at  the  point  (a?',  y')  on  the  curve 
is[(2),§85j 

2/-y=-£(^-^'),  (3) 

or     ^  2a(i/  — /)+2/'(^  — a?0  =  0.  (4) 

The  tangent  at  the  vertex  (0,  0)  is  the  line  a?  =  0;  and  the 
normal  at  the  same  point  is  2/  =  0,  i.  e.  the  axis  of  the  curve. 

Ex.  1.    Show  that  the  equation  of  the  parabola  is 

2/2  =  4a(x  rt  a), 

according  as  the  origin  is  at  the  focus  or  on  the  directrix. 

Ex.  2.    Change  the  equations  of  the  parabolas 

(y  — fc)2  =  4a(a;  — /i)    and    (x  — /i)^  =4a(2/  — fc) 

to  the  standard  form,  and  show  that  their  vertices  are  at  the  point  (ft,  fc). 

Ex.  3.    What  relation  does  the  line  (3)  have  to  the  parabola  when  the 
I)oint  (x^,  2/')  is  not  on  the  curve? 


188 


THE    PARABOLA. 


[124. 


1 24.     Geometric  properties  of  the  parabola. 


M 
R 

Y 

'.-^^ 

\ 

^ 

\. 

T                                 D 

O 

If                         no 

Let  the  tangent  at  the  point  P{qc!,  y')  meet  the  axis  in  T,  the 
directrix  in  B,  and  the  tangent  at  the  vertex  in  Q.  Let  PMand 
PN  be  the  perpendiculars  from  P  to  the  directrix  and  axis,  re- 
spectively. 

Let  the  normal  at  P  meet  the  axis  in  G, 

Then  we  have  the  following  properties : 

TO  =  ON=  x'.  [(2),  §  123.]          (1) 

.• .     Suhtangent  =  TN^20N  =  2x',  (2 ) 

Oq  =  lNP  =  \y'.  ^            (3) 

TF=FP=:FG  =  a-i-x\  (4) 

lFPR  =  /_MPR.  (5) 

Z  PFP  =  Z  BMP  =  Jtt.     (See  Ex.  39,  p.  178. )          (6) 

FM  is  perpendicular  to  TP.  (7) 

PM,  PT,  and  OY  meet  in  a  point.  (8) 

0G^2a-^x'.  [(4),  §123.]          (9) 

,*.     Subnormal  =  NG  =  2a,  a  constant.  (10) 


125.]  THE   PARABOLA.  189 

The  use  of  parabolic  reflectors  depends  on  the  property  ex- 
pressed in  (5).     Let  the  student  explain. 

Properties  (6)  and  (7)  suggest  a  method  of  drawing  tangents 
from  an  exterior  point.     Show  how  this  can  be  done. 

125.     Equations  of  the  tangent  and  normal  in  terms  of  the  slope  m. 
The  equation  of  the  tangent  [(2),  §  123]  may  be  written 

y  =  yx+^.  (2) 

Let  —r  =  m:  then  -^  =  -,  and  (2)  may  be  written 
i/  2        m  K   y        J 

y  =  mx-\--,  (3) 

which  is  the  required  equation.     That  is,  the  line  (3)  will  touch 
the  parabola  y^  =  4:ax,  whatever  the  value  of  m  may  be. 

In  a  similar  manner  it  can  be  shown  from  (3),  §  123,  that  the 
equation  of  the  normal  expressed  in  terms  of  its  slope  is 

y  =  mx  —  2am  —  am^.  (4) 

Ex.  Show  that  the  tangents  from  the  point  (x\  y^)  to  the  parabola  will 
be  real,  coincident,  or  imaginary  according  as  y^^  —  4aaj^>,   =,  or  <  0. 

EXAMPLES. 

1.  Find  the  equations  of  the  tangents,  and  the  normals  at  the  ends  of 
the  latus  rectum. 

2.  Find  the  value  of  a  if  the  parabola  y"^  =  iax  goes  through  (3,  2) ; 
(9,  —  12) .    How  many  conditions  can  the  curve  i/^  =  4ax  be  made  to  satisfy? 

3.  Show  that  the  line  y  =  3x-\-  q-  touches  the  parabola  y-  =  4ax;  and 
also  that  y  =  ix-\-^  touches  y^  =  Sax. 

4.  Find  the  equation  of  the  tangent  to  y"^  =  12a;  which  makes  an  angle 
of  60°  with  the  x-axis. 

5.  Find  the  equations  of  the  tangents  drawn  from  the  point  (—  2,  2)  to 
the  parabola  y^  =  6a;. 


190  THE    PARABOLA.  [125. 

Find  the  coordinates  of  the  vertex,  of  the  focus,  the  length  of  the  latus 
rectum,  and  the  equation  of  the  directrix  of  each  of  the  following  parabolas : 
6.    y'  =  Sx  +  Q.  7.    x2  +  4a;  +  22/  =  0.  8.     (2/  — 4)^  =  6(a;  +  2). 

9.    4(x  — 3^  =  3(2/ +  1).  10.    y'  +  8x-6y-{-i  =  0. 

11.  For  what  point  on  the  parabola  y^  =  iax  is  (1)  the  subtangent  equal 
to  the  subnormal,  and  (2)  the  normal  equal  to  the  difference  between  the 
subtangent  and  the  subnormal  ? 

12.  Show  that  the  lines  y=zt(x-\-  2a)  touch  both  the  parabola  y'^=  Sax 
and  the  circle  x'^-\-y^  =  2a^.     . 

13.  Find  the  equation  of  the  common  tangent  to  the  parabolas  y^  =  Aax 
and  x^  =  ihy.  Show  also  that  if  a  =  6,  the  line  touches  both  at  the  end  of 
the  latus  rectum. 

14.  Find  the  equations  of  the  tangents  to  the  parabolas  in  examples 
6,  7,  8,  9,  10  whose  slope  is  —  2. 

15.  Show  that  for  all  values  of  m  the  line 

2/  =  m(x  +  a)+—   will  touch  y^  =  ia(x -\- a) ; 

y  =  m{x  —  a)-\ —  will  touch  y'^  =  4:a{x  —  a); 
and         (y  —  k)=  m(x  —  ]i)-\-—   will  touch   (y  —  ky  =  4a(x  —  h) . 

16.  If  {x\  2/')  and  (x^^,  y^^)  are  the  points  of  contact  of  two  tangents 
to  2/^  =  4ax,  show  that  the  coordinates  of  their  point  of  intersection  are 

17.  Show  that  the  directrix  is  the  locus  of  the  vertex  of  a  right  angle 
whose  sides  slide  upon  a  parabola.     (§  125.) 

18.  Two  lines  are  perpendicular  to  one  another;  one  of  them  is  tan- 
gent to  2/^  =  4ta{x  +  ct),  and  the  other  is  tangent  to  y^  =  4b(x  +  6) ;  show 
that  these  lines  intersect  on  the  line  x -{- a -\- b  =  i). 

19.  Show  that  the  line  Ix  -\-  my  -f  n  =  0  will  touch  the  parabola  2/^  =  4aa;, 
if  In  =  am^. 

20.  If  the  chord  PQR  passes  through  a  fixed  point  Q  on  the  axis  of  the 
parabola,  show  that  the  product  of  the  ordinates,  and  also  the  product  of 
the  abscissas  of  the  points  P  and  R,  is  constant. 

21.  Find  the  coordinates  of  the  point  of  intersection  of  y  =  'mx-\ — 

and  2/  =  m^x  -{ -.    Show  that  the  locus  of  this  point  is  a  straight  line  if 

mm^  is  constant.    What  is  the  locus  when  mm^=  —  1  ? 

22.  If  perpendiculars  be  let  fall  on  any  tangent  to  a  parabola  from  two 
points  on  the  axis  which  are  equidistant  from  the  focus,  the  difference  of 
their  squares  will  be  constant. 

23.  All  chords  of  a  parabola  which  subtend  a  right  angle  at  the  vertex 
meet  the  axis  in  the  same  point. 


125.]  THE   PARABOLA.  191 

24.  The  vertex  ^  of  a  parabola  is  joined  to  any  point  P  on  the  curve, 
and  PQ  is  drawn  at  right  angles  to  ^P  to  meet  the  axis  in  Q.  Prove  that 
the  projection  of  PQ  on  the  axis  is  always  equal  to  the  latus  rectum. 

25.  If  P,  Q,  and  R  be  three  points  on  a  parabola  whose  ordinates  are  in 
geometrical  progression,  the  tangents  at  P  and  R  will  meet  on  the  ordinate 
of  Q. 

26.  Show  that  the  locus  of  the  intersection  of  two  tangents  to  a  par- 
abola at  points  on  the  curve  whose  ordinates  are  in  a  constant  ratio  is  a 
parabola. 

27.  Prove  that  the  circle  described  on  a  focal  radius  as  diameter  touches 
the  tangent  drawn  through  the  vertex. 

28.  Prove  that  the  circle  described  on  a  focal  chord  as  diameter  touches 
the  directrix. 

29.  Find  the  locus  of  the  point  of  intersection  of  two  tangents  to  a 
parabola  which  make  a  given  angle  a  with  one  another. 

If  a  =  45°,  show  that  the  locus  is 

y^  —  4ax  =  {x-\-  ay. 
If  a  =  60°,  show  that  the  locus  is 

2/2  _  33.2  _  loax  —  3a2  =  0. 

^Suggestion.    The    line    y  =  mx-\-—     will    go    through    {x^,  y^)    if 

m-x^  —  9711/''  -|-  a  =  0.    The  roots  of  this  equation  are  the  slopes  of  the  two 
tangents  which  meet  in  {x\  y^).    Let  mi,  m^  be  these  roots,  then  see  §  91. J 

30.  The  two  tangents  from  a  point  P  to  the  parabola  y^  =  4ax  make 
angles  tan"*  m-i  and  tan~^  m2  with  the  x-axis.  Find  the  locus  of  P,  (1)  when 
mi  -f  W2  is  constant,  (2)  when  mi^-{-m2^  is  constant,  and  (3)  when  mim^  is 
constant. 

31.  If  ^  is  the  area  of  a  triangle  inscribed  in  the  parabola  y^  =  4aa;,  and 
K^  is  the  area  of  the  triangle  formed  by  the  tangents  at  the  vertices  of  the 
inscribed  triangle,  prove  that 

where  2/1 1 2/21  ys  are  the  ordinates  of  the  vertices  of  the  inscribed  triangle. 
(See  Ex.  16.) 

Find  the  locus  of  the  middle  points 

32.  Of  all  ordinates  of  a  parabola. 

33.  Of  all  focal  radii. 

34.  Of  all  chords  through  the  fixed  point  (ft,  fc). 

As  special  cases,  let  (ft,  k)  be  (1)  the  focus,  (2)  the  vertex,  (3)  the  point 
(4a,  0),  and  (4)  the  point  ( — a,  0). 

35.    Show  that  the  parabola  is  concave  towards  its  axis. 


192 


THE    PARABOLA, 


[126. 


1 26.     The  locus  of  the  middle  points  of  a  system  of  parallel  chords  of 
a  parabola  is  a  straight  line  parallel  to  the  axis  of  the  parabola. 


Let  AB  be  any  one  of  the  chords,  let  P^(^x^,  y')  be  its  middle 
point,  and  let  y  be  the  angle  it  makes  with  the  axis  of  the  parabola. 
Then  the  equation  of  AB  may  be  written  [(4),  §  46] 


X  — X' 


y  —  y 


=  r, 


(1) 

(2) 


(3) 


cos  y  sm  y 

or  x  =  x'-\-r  cos  y,     y  =  y'-\-r  sin  y. 

Let  the  equation  of  the  parabola  be 

Substituting  in  (3)  the  values  of  x  and  y  given  by  (2),  we  have 
for  the  points  common  to  the  chord  and  the  curve 

(y+  r  sin  yY  =  4a(£c'+  r  cos  y), 

or         r^  sin V +  2(2/' sin  ^  —  2a  cos  y)r -^  y'^  —  4:ax'=0,         (4) 

a  quadratic  equation  in  r,  whose  roots  are  represented  by  the  dis- 
tances P'B  and  P'J..  Since  P'  is  the  middle  point  of  AB,  the 
sum  of  these  roots  is  zero.     That  is, 

y'  sin  y  —  2a  cos  y  =  0.  (§  91. ) 

2a 
Whence  t/'=  2a  cot  r  =  tt"?  (5) 


m 


where  m  is  the  constant  slope  of  the  chords. 


127.]  THE    PARABOLA.  193 

The  coordinates  of  P'  therefore  satisfy  the  equation 

2/  =  ^  =  2acotr.  (6) 

Hence  the  locus  of  P',  as  AB  moves  keeping  m  constant,  is  a 
straight  line  OX'  parallel  to  the  axis  of  the  parabola. 

Definition.  The  locus  of  the  middle  points  of  a  system  of 
parallel  chords  of  a  conic  is  called  a  Diameter;  and  the  chords 
it  bisects  are  oblique  double  ordinates  to  that  diameter  considered 
as  an  axis  of  abscissas. 

We  have  seen  in  §  112  that  a  diameter  of  a  parabola  meets  the 
curve  in  only  one  point  at  a  finite  distance  from  the  directrix. 
This  point  is  called  the  Extremity  of  the  diameter. 

Cor.     The  line  (6)  meets  the  curve  in  0'  where 

a:=±,  =  Ra,     y=~.  (7) 

The  equation  of  the  tangent  at  0'  is,  therefore  [(2),  123], 

Hence  the  tangent  at  the  extremity  of  a  diameter  is  parallel  to  the 
chords  bisected  by  that  diameter. 

127.  To  find  the  equation  of  a  parabola  when  the  axes  are  any 
diameter  and  the  tangent  at  its  extremity. 

Using  the  figure  of  §  126,  and  keeping  the  same  notation,  we 
will  let  O'P'  =  X,  the  new  abscissa,  and  P'B  =  y,  the  new  ordinate. 

Then  y  is  always  the  same  as  r  of  equation  (4),  §  126.  And 
since  the  coefficient  of  the  first  power  of  r  in  this  equation  is  zero, 
we  have 

^         sinV     V  ^^ 

2a 
where  .V'=— .  [(5),  §  126] 

and  sd=  R0'+  0'P'=  ~ -\- x,     [(7),  §  126] 

m 

^..     ^^_i^a:.  (2) 

14 


194  THE    PARABOLA.  [128. 

Now      FO'  =  a-^Ra  [(4),  §124] 

^a(l  +m^)^^l+tanV^      a 

m?  tanV  sinV* 

Therefore,  if  a!  =   .  ^     =  FO,  the  required  equation  is 

'if=^4:a'x.  (4) 

Hence  the  equation  if  =  4:ax  always  represents  a  parabola,  the 
ic-axis  being  a  diameter,  the  ^/-axis  the  tangent  at  its  extremity, 
a  the  distance  from  the  focus  to  the  origin,  and  4a  the  length  of 
the  focal  chord  parallel  to  the  ?/-axis. 

Formula  (6),  §  111,  by  means  of  which  equation  (2),  §  123,  was 
obtained,  and  also  the  derivation  of  equation  (3),  §125,  from 
equation  (2),  §  123,  hold  good  equally  whether  the  axes  are  rect- 
angular or  not.  That  is,'  if  the  equation  of  a  parabola  is 
y^  =  Aax,  the  line 

yi/  =  2a(ix  +  af)  (5) 

will  be  the  tangent  at  the  point  (x',  y')  if  the  point  is  on  the 
curve;  but  always  the  polar  of  (a/,  f/)  with  respect  to  the  par- 
abola.    And  the  line 

2/  =  ^^  +  ^  (6) 

will  also  touch  the  parabola  for  all  values  of  ?w-,  the  meaning  of  m 
being  that  given  in  §  59. 

CoR.  The  polar  of  any  point  with  respect  to  a  parabola  is  parallel 
to  the  chords  bisected  by  the  diameter  through  the  point. 

Conversely,  the  locus  of  the  poles  of  parallel  chords  is  the  bisecting 
diameter. 

For  the  polar  of  any  point  (.t',  0)  is,  by  (5),  x=^  — x'. 

1 28.     Through  any  point  three  normals  can  be  drawn  to  a  parabola. 

The  equation  of  the  normal  at  any  point  (xf,  2/)  of  the  par- 
abola y^  =--  4:ax  is  [(4),  §  123] 

M2/-y)+2/'(^-^')=0.  (1) 

If  the  line  (1)  goes  through  the  point  (^,  k),  then,  since 


128.] 


THE   PARABOLA. 


195 


we  have 


2a(^-2/)  +  2/(A-Q=0, 


or  y'^-\-^a(2ak  —  h)y'—Sa'k  =  0.  (2) 

The  three  roots  of  equation  (2)  are  the  ordinates  of  the  three 
points  the  normals  at  which  pass  through  any  given  point  (h,  k). 

Let  ?/i,  ijo,  ys  be  the  three  roots  of  (2)  ;  then,  since  the  coeffi- 
cient of  i/Ms  zero  (§  91), 

2/1  +  2/2  +  ^3  =  0.  (3) 


Let  yi  and  3/2  ^  ^^^  ordinates  of  the  ends  of  any  one,  AB,  of 
a  system  of  parallel  chords  whose  slope  is  m ;  then 

i(Z/i+2/2)  = 


m 


4a 


1/3  = ,  a  constant. 


[(6),  §126.]     (4) 
(5) 


Therefore,  the  normals  at  A  and  B,  as  AB  moves  keeping  m 
constant,  always  meet  on  the  fixed  normal  at  C  whose  ordinate 
4a 


IS 


m 


That  is,  the  locus  of  the  intersection  of  normals  at  the  ends  of  a  sys- 
tern  of  parallel  chords  of  a  parabola  is  the  normal  to  the  curve  at  the 
point  whose  ordinate  is  minus  twice  the  ordinate  of  the  middle  points  of 
the  chords. 


196  THE    PARABOLA.  [128. 


Examples  on  Chapter  VIII. 

1.  Find  the  equation  of  that  chord  of  the  parabola  y"^  —  6a;  which  is  bi- 
sected by  the  point  (4,  3). 

2.  Find  the  equation  of  the  chord  of  x^  =  —  %y  whose  middle  point  is 

(-3,-2). 

3.  Find  the  equation  of  a  normal  to  y"^  =  ix  which  shall  pass  through 
(2,-8;. 

4.  Find  the  equations  of  the  normals  to  the  parabola  y"^  =  8x  which  pass 
through  the  point  (8,  2). 

5.  Find  the  equations  of  the  normals  to  y"^  =  ^ax  which  meet  in  the  point 
(5,  0).    For  what  values  of  h  are  the  three  normals  real  ? 

6.  Show  that  the  axis  of  the  parabola  y"^  =  %x  divides  each  of  the  chords 

1/  X  ~\-  2 

whose  equations  are    .    ^r.^  =      ~^^o  into  two  segments  whose  product 
sin  ii\)       cos  ov 

is  64. 

7.  Any  tangent  to  a  parabola  will  meet  the  directrix  and  the  latus 
rectum  (produced)  in  two  points  equidistant  from  the  focus. 

8.  The  angle  between  two  tangents  to  a  parabola  is  equal  to  half  the 
angle  between  the  focal  radii  of  their  points  of  contact. 

9.  If  a  line  is  a  normal  to  a  parabola  at  one  end  of  the  latus  rectum,  its 
pole  with  respect  to  the  parabola  lies  on  the  diameter  through  the  other 
end  of  the  latus  rectum. 

10.  Show  that  the  locus  of  the  centre  of  a  circle  which  intercepts  a 

chord  of  given  length  2a  on  the  x-axis  and  passes  through  the  fixed  point 

<0,  5)  is  the  curve 

x'  —  2by-\-b'  =  a\ 

11.  Find  the  locus  of  the  centre  of  a  circle  which  touches  a  given  circle 
and  also  a  given  straight  line. 

12.  The  perpendicular  from  a  point  Q  on  its  polar  with  respect  to  a 
parabola  meets  the  polar  in  M  and  the  axis  in  G ;  the  polar  cuts  the  axis  in 
T,  and  the  ordinate  through  Q  meets  the  curve  in  P  and  P^.  Show  that  the 
points  r,  P,  ikf,  G,  P^  are  all  on  a  circle  whose  centre  is  F. 

13.  Prove  that  the  two  parabolas  y"^  =  ax  and  x^  =  by  cut  one  another  an 

angle  ^  tan  ^ ^ ^ 

2(a^+b^) 

14.  Show  that  the  two  parabolas 

a;2  +  4a(2/  — 26  — a)  =  0    and    y^  =  4b{x  —  2a-\-b) 
intersect  at  right  angles  at  a  common  end  of  a  latus  rectum  of  each. 


128.]  THE    PARABOLA.  197 

15.  If  EFG  is  a  focal  chord  of  a  parabola  whose  vertex  is  A,  and  GA 
meets  the  directrix  in  B,  show  that  BE  is  parallel  to  the  axis  of  the  par- 
abola. 

16.  Show  that  the  equation  of  the  chord  of  y"^  =  Adx  which  is  bisected 
at  the  point  (h.  k)  is 

fc(y  — fc)=2a(x  — /i). 

17.  Show  that  if  three  normals  meet  in  a  point,  the  sum  of  their  slopes 
is  zero.  Show  also  that  if  the  sum  of  the  slopes  of  two  normals  is  con- 
stant, the  locus  of  their  intersection  is  a  third  normal  to  the  parabola. 
[(4),  §125  and  §128.]  • 

18.  Show  that  the  locus  of  the  point  of  intersection  of  two  perpendicu- 
lar normals  to  the  parabola  y"^  =  iax  is  the  parabola  j/^  =  a{x  —  3a). 

19.  Show  that  if  two  tangents  to  a  parabola  intercept  a  fixed  length  on 
the  tangent  at  the  vertex,  the  locus  of  their  point  of  intersection  is  an- 
other equal  parabola. 

20.  The  tangents  and  the  normals  at  the  ends  of  any  focal  chord  inter- 
sect on  the  circle  whose  diameter  is  the  chord. 

21.  Show  that  two  tangents  to  a  parabola  which  make  complementary- 
angles  with  the  axis,  but  are  not  at  right  angles,  meet  on  the  latus  rectum. 

22.  A  perpendicular  drawn  from  the  vertex  of  the  parabola  y^  =  4ax  to 
the  tangent  at  any  point  P  meets  the  diameter  through  P  in  Q,  the  tan- 
gent in  JB  and  the  ordinate  through  P  in  S.  Show  that  the  loci  of  Q,  R, 
and  iS»  are,  respectively, 

a;  +  2a  =  0,    x^-{-  y'\x  -f  a)  =  0,    and    a^  =  ay^» 

(Draw  the  normal  at  P  and  the  ordinate  of  Q.) 

23.  From  any  point  on  the  latus  rectum  of  a  parabola  perpendiculars 
are  drawn  to  the  tangents  at  its  extremities.  Show  that  the  line  joining 
the  feet  of  these  perpendiculars  touches  the  parabola. 

24.  Show  that  the  locus  of  the  foot  of  the  perpendicular  drawn  from 
the  focus  to  any  normal  to  the  parabola  y"^  =  4aaj  is  the  parabola 


25.  Show  that  if  tangents  are  drawn  to  the  parabola  y^  =  iax  from  any 
point  on  the  line  x-\~^a  =  0,  their  chord  of  contact  will  subtend  a  right 
angle  at  the  vertex. 

26.  Prove  that  the  chord  of  the  parabola  y"^  =  4ax,  whose  equation  is 
y  —  xl/2  +  4a/2  =  0,  is  a  normal  to  the  curve  and  that  its  length  is  6v^3a. 

27.  The  perpendicular  TN  from  any  point  T  on  its  polar  with  respect  to 
a  parabola  meets  the  axis  in  M,  Show  that  if  TN  •  TM  is  constant,  or  if 
the  ratio  TN :  TM  is  constant,  the  locus  of  T  is  a  parabola. 


198  THE    PARABOLA.  [128. 

28.  Two  equal  parabolas  have  their  axes  parallel  and  a  common  tangent 
at  their  vertices;  a  straight  line  is  drawn  parallel  to  their  axes  meeting  the 
parabolas  in  P  and  Q.  Show  that  the  locus  of  the  middle  point  of  PQ  is  an 
equal  parabola. 

29.  Two  parabolas  have  a  common  axis  and  concavities  in  opposite  di- 
rections; if  any  line  parallel  to  the  common  axis  meets  the  curves  in  Pand 
Q,  prove  that  the  locus  of  the  middle  point  of  PQ  is  another  parabola,  pro- 
vided the  given  parabolas  are  not  equal. 

30.  Two  parabolas  touch  one  another*  and  have  their  axes  parallel. 
Shov/  that,  if  the  tangents  at  two  points  of  these  parabolas  meet  in  any 
point  on  their  common  tangent,  the  line  joining  the  points  of  contact  will 
be  parallel  to  their  axes. 

31.  Two  parabolas  have  the  same  axis.  Find  the  locus  of  the  middle 
points  of  chords  of  one  which  touch  the  other.  ^ 

32.  Two  parabolas  have  the  same  axis;  tangents  are  drawn  from  points 
on  the  first  to  the  second ;  prove  that  the  middle  points  of  the  chords  of 
contact  with  the  second  lie  on  a  fixed  parabola. 

33.  Two  parabolas  have  a  common  focus,  and  their  axes  in  opposite  di- 
rections. Prove  that  the  locus  of  the  middle  points  of  chords  of  either 
which  touch  the  other  is  another  parabola. 

34.  Two  equal  parabolas,  A  and  5,  have  the  same  vertex  and  their  axes 
in  opposite  directions.  Prove  that  the  locus  of  the  poles  with  respect  to  B 
of  tangents  to  A  is  the  parabola  A. 

35.  Show  that  the  locus  of  the  poles  of  tangents  to  the  parabola  j/'^  =  iax 
with  respect  to  the  parabola  y^  —  45x  is  the  parabola  ay^  —  W^x, 

36.  The  locus  of  the  poles  of  tangents  to  either  of  the  parabolas 
^y^  —  4ax  ov  x^  —  —  ^.ay  with  respect  to  the  other  is  xy  =  2a^. 

37.  If  a  line  touches  the  circle  x^-'ry^  =  4a^  its  pole  with  respect  to  the 
parabola  y"^  =  iax  lies  on  the  rectangular  hyperbola  x^  —  y'^  =  4a^ 

38.  The  middle  point  of  a  chord  PQ  is  on  a  fixed  straight  line  perpen- 
dicular to  the  axis  of  a  parabola;  show  that  the  locus  of  the  pole  of  the 
chord  is  another  parabola. 

39.  The  base  of  a  triangle  is  2a,  and  the  sum  of  the  tangents  of  the 

base  angles  is  k.    Show  that  the  locus  of  the  vertex  is  a  parabola  whose 

2a 
latus  rectum  is  —r-. 

40.  If,  in  the  triangle  ABC,  AB  is  constant  and  tan  A  tan  ^B  =  2,  the 
locus  of  C  is  a  parabola  of  which  A  is  the  vertex  and  B  is  the  focus. 


128.]  THE    PARABOLA.  199 

41.  If  ^  is  the  angle  which  a  focal  chord  makes  with  the  axis,  prove  that 
the  length  of  the  chord  is  4a  csc''  6,  and  the  length  of  the  perpendicular  on 
it  from  the  vertex  is  a  sin  0. 

42.  Two  parallel  chords  of  a  parabola  meet  the  axis  in  points  equidis- 
tant from  the  vertex.  Show  that  the  axis  divides  each  chord  into  two  seg- 
ments whose  products  are  equal.    (Use  (4),  §  46.) 

43.  PQ  is  any  one  of  a  system  of  parallel  chords  of  «a  parabola;  O  is 
any  point  on  PQ  such  that  the  product  PO  •  OQ  is  constant.  Show  that  the 
locus  of  O  is  a  parabola. 

44.  Prove  that  the  locus  of  the  middle  point  of  that  portion  of  the  nor- 
mal intercepted  between  the  curve  y'^  =  4ax  and  its  axis  is  a  parabola  whose 
vertex  is  the  focus  and  whose  latus  rectum  is  a. 

45.  The  locus  of  the  middle  points  of  normal  chords  of  the  parabola 
y"^  =  4ax  is 

y*  —  2ay\x  —  2a)  +  8a*  =  0. 

46.  Prove  that  the  distance  between  a  tangent  to  the  parabola  y"^  =  4ax 
and  the  parallel  normal  is  a  esc  0  sec^  6^  where  d  is  the  angle  that  either 
makes  with  the  axis. 

47.  If  the  normals  at  two  points  of  a  parabola  are  inclined  to  the  axis 
at  angles  ^  and  (f>  such  that  tan  0  tan  ^  =  2,  show  that  they  intersect  on  the 
parabola.  \ 

48.  The  locus  of  a  point  from  which  two  normals  can  be  drawn  making 
complementary  angles  with  the  axis  is  a  parabola. 

49.  Two  equal  parabolas  have  the  same  focus  and  their  axes  are  at  right 
angles;  a  normal  to  the  one  is  perpendicular  to  a  normal  to  the  other; 
prove  that  the  locus  of  the  intersection  of  these  normals  is  another  par- 
abola. 

50.  If  a  normal  to  a  parabola  makes  an  angle  (p  with  the  axis,  show  that 
it  will  cut  the  curve  again  at  an  angle  tan-^(|  tan  (p). 

51.  If  TPand  TQ  are  tangents  to  a  parabola  whose  vertex  is  A,  and  if 
the  lines  PA,  QA,  TA,  produced  if  necessary,  meet  the  directrix  in  P^,  Q^,  T', 
respectively,  show  that  P^T^=  T'Q\ 

52.  Prove  that  there  is  a  fixed  point  K  on  the  axis  of  any  parabola  such 
that 

PK^^  QK' 

is  the  same  for  all  positions  of  the  chord  PKQ. 

53.  If  the  diameter  through  any  point  O  on  a  paraoola  meets  any  chord 
in  P,  and  the  tangents  at  the  ends  of  that  chord  in  Q  and  R,  show  that 

op'  =  oqoR. 


200  THE    PARABOLA.  [128. 

54.  A  chord  is  normal  to  a  parabola  and  makes  an  angle  0  with  the  axis. 
Prove  that  the  area  of  the  triangle  formed  by  it  and  the  tangents  at  its 
extremities  is  4a^  sec^  6  csc^  6. 

55.  The  vertex  of  a  triangle  is  fixed,  the  base  is  of  constant  length  and 
moves  along  a  fixed  straight  line.  Show  that  the  locus  of  the  centre  of  its 
circumscribing  circle  is  a  parabola. 

56.  A  chord  of  the  parabola  y^  —  iax  passes  through  the  fixed  point 
( —  2a,  0).    Prove  that  the  normals  at  its  extremities  meet  on  the  curve. 

57.  If  from  any  point  on  a  focal  chord  of  a  parabola  two  tangents  are 
drawn,  these  two  tangents  are  equally  inclined  to  the  tangents  at  the  ends 
of  the  chord. 

58.  If  Ti  and  r-i  are  the  lengths  of  radii  vectores  of  a  parabola  which  are 
drawn  at  right  angles  to  one  another  from  the  vertex,  prove  that 

ri^r2S  =  16aXri3  -\-  Vo's), 

59.  On  the  diameter  through  a  point  O  on  a  parabola  two  points  P  and 
Q  are  taken  such  that  OP  •  OQ  is  constant ;  prove  that  the  four  points  of 
intersection  of  the  tangents  drawn  fromP  and  Q  will  lie  on  two  fixed 
straight  lines  parallel  to  the  tangent  at  O  and  equidistant  from  it. 

60.  Tis  the  pole  of  the  chord  PQ;  prove  that  the  perpendiculars  from 
P,  r,  and  Q  on  any  tangent  to  the  parabola  are  in  geometrical  progression. 

61.  PFQ  is  a  focal  chord  of  a  parabola;  E  is  the  middle  point  of  PQ, 
and  RO  is  perpendicular  to  PQ  and  meets  the  axis  in  O;  prove  that  FO  and 
RO  are  the  arithmetic  and  geometric  means  between  FP  and  FQ. 

62.  Prove  that  the  locus  of  the  point  of  intersection  of  two  tangents,, 
which  with  the  tangents  at  the  vertex  form  a  triangle  of  constant  area  c^, 
is  the  curve 

x^(2/^  —  4ax)  =  4a^c^. 

63.  Parallel  chords  are  drawn  to  a  parabola;  the  locus  of  the  intersec- 
tion of  tangents  at  the  ends  of  these  chords  is  a  straight  line  (Cor.  §  127) ; 
and  the  locus  of  the  intersection  of  normals  at  these  points  is  also  a  straight 
line  (§  128).  Show  that  the  locus  of  the  intersection  of  these  two  lines, 
as  the  chords  change  direction,  is  a  parabola. 

64.  Show  that  the  locus  of  the  poles  of  chords  which  subtend  a  con- 
stant angle  a  at  the  vertex  is 

(x  +  4:a)2  =  4  cot^  a(2/2  —  4ax). 

65.  Prove  that  three  tangents  to  a  parabola,  which  are  such  that  the 
tangents  of  their  inclinations  to  the  axis  are  in  a  given  harmonical  pro- 
gression, form  a  triangle  whose  area  is  constant. 

66.  Find  the  equation  of  the  parabola  when  the  axes  are  the  tangents 
at  the  ends  of  the  latus  rectum. 


CHAPTER  IX. 

THE  CIRCLE. 

129.  Equations  of  the  circle,  and  the  corresponding  equations  of 
the  tangent,  polar,  and  normal. 

We  have  seen  in  §  82  that  the  equation  of  the  circle  whose 
radius  is  r  takes  the  simple  form 

x'-\-7/=r',  (1) 

when  the  origin  is  at  the  centre ;  while  if  the  centre  is  at  th« 
point  (a,  b)  the  equation  may  be  written 

(x  —  ay-\-(y  —  by=r\  (2) 

Moreover,  it  was  further  shown  in  §  110  that  the  general  equa- 
tion of  the  second  degree  will  represent  a  circle  if  a  =  6,  and 
h  =  0;  so  that  the  most  general  equation  of  a  circle  in  rectangular 
coordinates  is 

x'  +  2/^  +  2gx  H-  2/7/  +  c  =  0.  (3) 

Equation  (3)  may  be  put  in  the  form  (2),  which  gives 

(x+gy  +  (y+fy=9'+f-e.  (4) 

Hence  the  centre  of  the  circle  represented  by  (3)  is  the  point 
( —  g,  — /),  and  the  radius  is  equal  to  Vg'^  +/^  —  c. 

The  circle  will  therefore  be  real,  a  point,  or  imaginary  accord- 
ing as  ^r^ +/2_c  >,    =  ,    or  <0. 

By  applying  the  rule  of  §  111  to  equations  (1),  (2),  and  (3), 
respectively,  we  obtain 

xx'-^yy'=r','  (5) 

(x-a)(x'-a)  +  C!J-f>)(y-b)=r\  (6) 

and  xx'-{-  yy'+  g(x  +  x')+f(y  +  i/')+  c  =  0.  (7) 

These  are  the  equations  of  the  tangent  to  the  circles  (1),  (2), 
(3),  respectively,  at  the  point  (x',  y')  if  this  point  is  on  the 
curve;  but,  by  §  113,  they  are  always  the  equations  of  the  polar 
of  the  point  {x',  y')  with  respect  to  the  circles  represented  by 
(1),  (2),  (3). 


202  THE    CIRCLE.  [129. 

Since  the  normal  (§  78)  at  any  point  (x',  y')  of  the  circle 
x^  -\-  qf^r^  is  perpendicular  to  (5),  its  equation  is  [(2),  §  85] 

y  —  y=^(^  —  ^)y 

or  xy' — x'y  =  0.  (8) 

That  is,  the  normal  at  any  point  of  a  circle  passes  through  the 
centre. 

The  equations  of  the  normals  bo  the  circles  (2)  and  (3)  at  the 
point  (x'j  y')  are,  respectively  [(2),  §85], 

2/-^'=|E^(^-^'),  (9) 

and  2/-2/'  =  5t^(^-^');  (10) 


or 


xy'-x'y--h{x~x')^a{y-^y')=0,  (11) 


and  xy'—x'y  ^ fix  —  x')  —  g{y  —  y')=  0.  (12) 

The  general  equation  of  the  circle  (3),  or  (2),  contains  three 
parameters,  or  constants.  Therefore  a  circle  can  be  made  to  sat- 
isfy three  conditions,  and  no  more.  If  we  wish  to  find  the  equa- 
tion of  a  circle  which  satisfies  three  given  conditions,  we  assume 
the  equation  to  be  of  the  form  (3),  or  (2),  and  then  determine 
the  values  of  the  constants  ^,  /,  c,  or  a,  h,  r,  from  the  given  con- 
ditions. 

^^y  Find  the  equation  of  the  circle  passing  through  the  three  points 
(Ori),   (2,0),  and   (0,-3). 

Let  the  equation  of  the  required  circle  be 

a:^  +  2/^  +  2srx4-2/2/-fc  =  0.  (1) 

Since  the  given  points  are  on  the  circle,  their  coordinates  must  satisfy 
equation  (1). 

.-.    l-l-2/-fc  =  0,    4  +  4p  +  c  =  0,    9-6/+c  =  0. 

Whence  we  find  g  =  —  j,  /=  1,  and  c  =  —  3.  Substituting  these  values 
in  (1)  the  required  equation  becomes 

x'-\-y^~ix-]-2y  —  S  =  0. 
The  centre  is  the  point  (i,  —  1),  and  the  radius  is  ^V^. 


130.] 


THE  CIRCLE. 


203 


130.     A  geometrical  comstruction  for  the  polar  of  a  point  with  re- 
spect to  a  circle. 


Let  the  equation  of  the  circle  be 

x'-^f  =  r\  (1) 

Let  P(x',  y')  be  any  point,  BC  its  polar,  and  let  OP  and  BC 
intersect  in  Q.     Then  the  equation  of  ^0  is  [(5),  §  129] 

xx'-\-yi/=r\  (2) 

and  the  equation  of  the  line  OP  is  (§47) 

xy'—x'y  =  0,  (3) 

Hence  J50is  perpendicular  to  OP  (§  48),  and  therefore 


0Q  = 


Vx'^  +  y' 


[(5),  §60.]      (4) 


OP  =  l/a;"  +  ij'". 

[(4), 

§7.]      (5) 

OP-OQ  =  r'. 

(6) 

Also 


We  therefore  have  the  following  construction  for  the  polar  of  a 
point  P.  Draw  OP  and  let  it  cut  the  circle  in  B ;  then  construct 
a  third  proportional,  OQ,  to  OP  and  r,  i.  e.  take  Q  on  the  line 
OP,  such  that  OPiOR  =  OEiOQ,  and  draw  a  line  through  Q 
perpendicular  to  OP. 

Ex.  1.     Construct  the  pole  of  a  given  line. 

Ex.  2.    Prove  the  theorem  of  §  115  for  the  circle. 


204  THE   CIRCLE.  [131. 

131.     To  find  the  equation  of  'the  tangent  to  the  circle 

x'-i-f  =  r'  (1) 

in  terms  of  its  slope  m. 

The  line 

y  —  mx-\-h  (2) 

will  touch  the  circle  (1)  if  the  perpendicular  distance  from  it  to 
the  origin  is  equal  to  the  radius  r  of  the  circle  ;  that  is,  (§  50)  if 

h  , 

or     b  =  rVl-{-m\  ^3) 


l/l  +  m^ ' 
Therefore  the  straight  line 


y  =  mx -\- TV  i -\- m^  (4) 

will  touch  the  circle  (1)  for  all  values  of  m. 

Since  either  sign  may  be  given  to  the  radical  \/l  -f  m^  in  (3), 
it  follows  that  there  are  two  tangents  to  the  circle  for  every  value 
of  m ;  i.  e.  there  are  two  tangents  parallel  to  any  given  straight 
line. 

Ex.  1.  Derive  equation  (3)  by  treating  (1)  and  (2)  simultaneously  and 
taking  the  condition  for  equal  roots. 

Ex.  2.    What  is  the  equation  of  the  normal  to  (1)  in  terms  of  its  slope  ? 

Ex.  3.    How  many  normals  can  be  drawn  from  a  point  to  a  circle? 


EXAMPLES. 
Find  the  centres  and  radii  of  the  following  circles : 
''l.;  x2  +  2/2±4x  =  0.  2.    x2  +  2/2  ± 61/ =  0. 

3.    x2+2/2-f  2x  — 42/=:0.  4.    x'' -{-y'-  —  3x-\-6y  =  0. 

^    a;2  +  t/2  +  6x  — 42/  +  9  =  0.  fQ  i(x'^y')—i2x  +  Sy-]-23  =  0, 

Find  the  equation  of  the  circle  passing  through  the  three  points 

7.    (0,0),  (6,0),  (0,4).  8.    (0,0),   (1,1),   (4,0). 

I?)  (2,-3),   (3,-4),   (-2,-1).       10.    (1,2),   (3,-4),   (5,6). 
11.    (0,0),   (a,0),   (0,6).  12.    (a,0),   (-a,0),   (0,-6). 

13.  Find  the  equation  of  a  circle  passing  through  (0,  4)  and  (6,  0),  and 
having  ^^13  for  radius. 

14.  Find  the  equation  of  a  circle  whose  centre  is  (3, 4). and  which  touches 
the  Una  4x  —  dy  +  20  =  0. 


131.]  THE   CIRCLE.  205 

15.  Find  the  general  equation  of  the  circle  which  touches  both  axes. 

16.  Find  the  equation  of  the  circle  passing  through  the  point  ( —  3,  6) 
and  touching  both  axes. 

17.  Find  the  equation  of  the  circle  touching  the  line  y  —  c  and  both  axes. 
Write  down  the  equation  of  the  tangent  to  the  circle 

18.  X2  +  2/2  — 2x  +  32/  — 4  =  0atthepoint  (2,  1). 

19.  x2  +  2/2_|_4a._62/_i3  =  0atthepoint  (— 3,  — 2). 


20.  Show  that  the  lines  y  =  m{x  —  r)±zrV\-\-m^  touch  the  circle 
x'^  -f  2/^  =  2rx,  whatever  the  value  of  m  may  be. 

Find  the  equations  of  the  tangents  to  the  circle 

21.  x2  +  2/2  =  4  parallel  to  2ic  +  32/  +  1  =  0. 

22.  a;2  +  2/2  =  6x  parallel  to  3x  —  22/  +  2  =  0. 

23.  9(a;2  +  2/')  —  9(6x  —  %y)  +  125  =  0  parallel  to  3x  +  42/  =  0. 

24.  Show  that  the  line  x  —  2y  =  0  touches  the  circle 

25'  +  2/'  —  4aj  +  82/  =  0. 

25.  The  line  2/  =  3a;  —  9  touches  the  circle 

x2  +  2/2  +  2x  +  42/  — 5  =  0. 
Find  the  coordinates  of  the  point  of  contact. 

26.  Find  the  equation  of  the  tangent  to  x'  -f  2/"  =  ^^  (1)  which  is  per- 
pendicular to  y  =  mx-\-bj  (2)  which  passes  through  the  point  (c,  0), 
(3)  which  makes  with  the  axes  a  triangle  whose  area  is  r\ 

Find  the  polar  of  the  point 

27.  (1,2)  with  respect  to  x'  +  2/'  =  5. 

28.  (3,  —  2)  with  respect  to  3(x2  +  y^)=  14. 

29.  C— 4,  1)  with  respect  to*  x'^-f  2/2  — 2a; +  62/ +  7  =  0. 

30.  (2,  —  D  with  respect  to  x^  -f  2/^  +  3x  —  52/  +  3  =  0. 

31.  (—  a,  5)  with  respect  to  x^  +  2/2  —  2ax  +  2by  -\-a^  —  b^  =  0. 
Find  the  pole  of  the  line 

32.  2x-\-y  =  1  and  x  —  dy  —  1  with  respect  to  x'^-\-y'^  =  2. 

33.  x  —  2y  —  3  and  2x-\-y  =  4:  with  respect  to  x^  -\-  y'^  =  6. 

34.  X  +  2/  4- 1  =  0  with  respect  to  x^  +  2/^^  +  4x  —  62/  +  U  =  0. 

35.  2x  +  Uy  =  15  with  respect  to  2(x2 -\-y^)  —  Sx-\-5y  —  2  =  0. 

36.  3(ax  —  by)  =0^  +  6^  with  respect  to  x^  +  2/'  —  2ax  +  2by  =  a^-{-  6^ 

37.  Show  that  the  circles  x"^  -^  y'^  —  ix  -\- 2y  =  15  and  x^  _|_  ^^2  _  5  touch 
one  another  at  the  point  (—  2,  1). 


206  THE   CIRCLE.  [131. 

38.  Show  that  the  polars  of  the  point  (1,  0)  with  respect  to  the  two  cir- 
cles x^-^y'^-{-ix  —  14  =  0  and  x^  -f-  2/^  =  4  are  the  same  line ;  show  that  the 
same  is  true  of  the  point  (4,  0). 

39.  Find  two  points  such  that  the  polars  of  each  with  respect  to  the  two 
circles  x^-^-y"^  —  2x  —  3  =  0  and  x^-\-y'^-^2x  —  11  =  0  coincide. 

40.  A  certain  point  has  the  same  polar  with  respect  to  two  circles;, 
prove  that  any  common  tangent  subtends  a  right  angle  at  that  point. 
Show  also  that  there  are  two  such  points  for  any  two  circles. 

41.  Find  the  locus  of  the  intersection  of  two  tangents  to  x- -{- y"'' =  r^ 
which  are  at  right  angles  to  one  another. 

42.  Find  the  locus  of  the  intersection  of  two  tangents  to  x'^-\-y'^  =  r^ 
which  intersect  at  an  angle  a. 

43.  Show  that  if  the  coordinates  of  the  extremities  of  a  diameter  of  a 
circle  are  (xi,  2/1)  and  (a;2, 2/2)*  respectively,  the  equation  of  the  circle  will  be 

(x  — xi)(x  — a;2)  +  (2/  — 2/0(2/  — 2/2)  =0. 

[Suggestion.  Lines  joining  any  point  (x,  y)  on  the  circle  to  (xi,  yi)  and 
(X2, 2/2)  are  at  right  angles  to  one  another.] 

Find  the  equation  of  the  circle  which  touches 

44.  the  lines  x  =  0,  x  =  a,  and  Sy  =  ix-\-  3a. 

One  Ans.    4(ic2  +  y'')  —  4a(x  +  5?/)  -f-  250^  =  0. 

45.  both  axes  and  the  line — 1-|-  =  1. 

a       0 

46.  Prove  analytically  that  the  locus  of  the  middle  points  of  a  system 
of  parallel  chords  of  a  circle  is  the  diameter  perpendicular  to  the  chords. 
(See  §  126.) 

47.  Show  that  as  a  varies  the  locus  of  the  intersection  of  the  lines 
x  cos  a  -j-  2/  sin  a  =  a  and  x  sin  a  —  y  cos  a  =  6  is  a  circle. 

48.  A  circle  touches  the  y-axis  and  cuts  off  a  constant  length  (2a)  from 
the  X-axis;  show  that  the  locus  of  its  centre  is  x^  —  y^  =  a^. 

49.  Two  lines  are  drawn  through  the  points  (a,  0)  and  (—a,  0)  and 
make  an  angle  a  with  one  another.  Show  that  the  locus  of  their  point  of 
intersection  is 

x^  +  2/"  ±  2ay  cot  a  =  a^. 

50.  If  the  polar  of  the  point  (x%  y^ )  with  respect  to  the  circle  x'^-\-y'^  =  a' 
touches  the  circle  x^-\-y^  =  2ax,  show  that  y^^  -\-  2ax^=  a*. 

51.  Show  that  if  the  axes  are  inclined  at  an  angle  w,  the  equation  of  the 
circle  is  (§  8) 

(x  —  a)2  +  (2/ —  b)2 -f  2(x  —  a)(2/ —  6)  cos  w  =  r2, 

where  (a,  h)  is  the  centre  and  r  the  radius. 


132.] 


THE   CIRCLE. 


207 


1 32.     To  find  the  length  of  a  tangent  drawn  from  a  given  point 
P{x',  y')  to  a  given  circle. 

Let  the  equation  of  the  circle  be 

(a:-ay-^{y-by-r^  =  0.  (1) 

Let  C  be  the  centre  and  FT  one  tangent  from  P. 

Y 


(2) 


(3J 


Then,  since  CPT  is  a  right  triangle, 

PT'  =  CP'  —  CT\ 

But  CP'  =  {X'-  ay  +  (y'-  by,  [§  7,  (2)] 

and  CT'  =  rl 

.-.     PT'=  {x'—  ay  +  {y'—hy  —  r". 

That  is,  the  square  of  the  tangent  is  found  by  substituting  the 
coordinates  x',  y'  of  the  given  point  in  the  left  member  of  equa- 
tion (1). 

Since  the  general  equation  of  the  circle, 

x'J^y^^2gx-\-2fy-Vc^0,  (4) 

can  be  put  in  the  form  of  (1)  by  merely  adding  and  subtracting 
g^  and  /^  in  the  first  member,  it  follows  that  if  the  coordinates  of 
any  point  are  substituted  in  the  first  member  of  (4)  the  result 
will  be  equal  to  the  square  of  the  length  of  the  tangent  drawn 
from  the  point  to  the  circle ;  or  the  product  of  the  segments  of  any 
chord  {or  secant)  drawn  through  the  point. 

Ex.  1.    What  is  the  meaning  of  (3)  when  the  second  member  is  negative? 

Ex.  2.    "What  is  represented  by  c  in  equation  (4)? 

Ex.  3.    Where  is  the  origin  if  c  is  positive  ?  if  c  is  zero  ?  if  c  is  negative  ? 


208 


THE   CIRCLE. 


[133. 


1 33.  If  a  circle  passes  through  the  common  points  of  two  given  circles, 
tangents  drawn  from  any  point  on  it  to  the  two  given  circles  are  in  a 
eonstant  ratio. 


Let  S  =  x'-]-f-{-2gx-\-2fy-\-c  =  0  (1) 

and  S'=x'-\-y'-^2g'x-\-2f7j-{-c'=0,  (2) 

be  the  equations  of  the  two  given  circles. 

Then  the  locus  of  ^  =  XS',  i.  e.  (See  Ex.  10,  p.  71. ) 

x'+  f+2gx  ^2fij  -^  c  ^  X(x'+  f-\-2g'x-j-2f'y  -^  c'),     (3) 

for  all  values  of  A,  will  pass  through  the  common  points,  A,  B,of 
(1)  and  (2).  Moreover,  (3)  is  a  circle  (§  110),  and  therefore,  for 
different  values  of  A,  represents  all  circles  through  the  intersection 
of  (1)  and  (2). 

Let  P(x',  y')  be  any  point  on  (8);  let  PT  and  PT'  be  the  tan- 
gents to  (1)  and  (2)  respectively.  Then  the  coordinates  x',  y' 
must  satisfy  (3),  and  we  therefore  have 

x"+  2/''+  25f^'+  W+  c  =  Kx"^-y''+  2g'x'-\-2ry'-\-  c').    (4) 

Therefore  PT'=l'PT'\  (§132)     (5) 

which  proves  the  proposition,  since  A  is  constant  for  any  particu- 
lar circle. 


133.]  THE   CIRCLE.  209 

When  ^  =  1,  it  is  easy  to  show  that  the  radius  and  the  coordi- 
nates of  the  centre  (§  J29)  of  the  circle  represented  by  equation 
(3)  all  become  infinite.     In  this  case  the  equation  reduces  to 

2{g-g')x-\-2U-r)y-\-c-c'=0,  (6) 

which  is  of  the  first  degree,  and  therefore  represents  the  straight 
line  AB  through  the  common  points  of  the  two  given  circles. 

Let  QR  and  QR'  be  tangents  to  /S  =  0  and  /S'=0,  respectively, 
from  any  point  Q  on  AB\  then,  since  ABQ  is  the  circle  through 
the  common  points  of  (1)  and  (2)  corresponding  to  A  =  1,  it 
follows  from  (5)  that 

QR=QR'.  (7) 

That  is,  tangents  drawn  to  the  two  given  circles  from  any  point 
on  the  line  (6)  are  equal. 

It  is  to  be  noticed  that  the  straight  line  given  by  (6)  is  in  all 
cases  real,  provided  g,  /,  c,  g',  /',  c'  are  real,  although  the  circles 
/S  =  0  and  /S"=0  may  not  intersect  in  real  points;  in  fact  one  or 
both  of  the  circles  may  be  wholly  imaginary.  We  have  here, 
therefore,  the  case  of  a  real  straight  line  passing  through  the 
imaginary  points  of  intersection  of  two  real  or  imaginary  circles. 
(C/.§113.) 

Definition.  The  straight  line  through  the  points  of  intersec- 
tion (real  or  imaginary)  of  two  circles  is  called  the  Radical 
Axis  of  the  two  circles. 

From  equation  (7)  it  follows  that  the  radical  axis  may  also  be 
defined  as  the  locus  of  the  points  from  which  tangents  drawn  to 
the  two  circles  are  equal  to  one  another. 

CoR.  If  the  coefficients  of  x^  in  S  and  S'  are  unity,  the  equation  of 
the  radical  axis  of  the  two  circles  S  =  0  and  S'=0  is  S  —  S'=0. 

Ex.  1.  Show  that  the  radical  axis  of  two  circles  is  perpendicular  to  the 
line  joining  their  centres. 

Ex.  2.  If  tangents  are  drawn  to  two  circles  from  any  point  on  a  line 
parallel  to  their  radical  axis,  show  that  the  difference  of  the  squares  of 
these  tangents  is  constant. 

Ex.  3.    Show  that  the  radical  axis  of  two  circles  divides  the  line  joining 
their  centres  into  two  segments,  such  that  the  difference  of  their  squares 
is  equal  to  the  difference  of  the  squares  of  the  radii. 
15 


210  THE   CIRCLE.  [134. 

1 34.  The  radical  axes  of  three  circles,  taken  in  pairs,  meet  in  a 
point. 

Let/Si=0,  /S2=0,  ^^3=0  be  the  equations  of  three  circles,  in 
each  of  which  the  coefficient  of  x^  is  unity. 

Then  the  equations  of  their  three  radical  axes  are  (§  133,  Cor.) 

/S\  —  S2=0j     S2  —  ^^3=0,     S3  —  Si^O. 

The  sum  of  any  two  of  these  equations  is  equivalent  to  the 
third.  Hence  they  form  a  consistent  system,  and  therefore  their 
loci  meet  in  a  point.      (See  §  53,  Ex.) 

This  point  is  called  the  Radical  Centre  of  the  three  circles. 

EXAMPLES. 

Find  the  length  of  the  tangents  (or  the  product  of  the  segments  of  the 
chords)  drawn  from  the  points 

1.  (3,  2),  (5,  —  4)  to  the  circle  x^  +  2/^  =  4. 

2.  (—3,  2),  (4,  —  4)  to  the  circle  x^  + 1/'  =  25. 

3.  (3,  —  2) ,  (1,  3)  to  the  circle  x^  +  2/^  —  2x  —  Ay  =  0. 

4.  (2,  1),  (0,  0)  to  the  circle  2{x'  +  y')  —  12x  —  4i/  +  15  =  0. 
-       5.     (0,  0),(— 2,  —  5)  tothecirclex2  +  2/2  — 6xH-42/  +  4  =  0. 

6.  (0,  0),  (6, —3)  to  the  circle  x2  + 2/2  + 6x  — 8?/  — 11=0. 

Find  the  radical  axis  of  the  circles 

7.  x2  +  2/'  +  6x  — 42/  — 3  =  0  and  x2  +  2/'  — 4x  +  82/  — 5  =  0. 

8.  x2-f  2/'  — 8x  — 102/  +  25  =  0  and  x''-\-y'' -\-8x  —  2y  +  8  =  0. 

9.  x"^ -\- y^ -{- ax -\- by  —  c  =  0  and  ax^  +  ay^  -\-  a^x  +  b^y  =  0. 

10.  Find  the  radical  axis  and  the  length  of  the  common  chord  of  the 
circles 

x^  -\-  y^  -\-  ax  -\-hy  -\-  c  =  Q  and  x"^  -\-  y"^  -\-h:p  -\-  ay  -\-  c  =  0. 

11.  Show  that  the  three  circles 

a;2  -)-  2/2  —  2x  —  42/  =  0,  x^  +  2/'  —  6x  +  42/  +  4  =  0, 

x2  +  2/'  — 8x  +  8y  +  6  =  0 
have  a  common  radical  axis.    Find  the  equation  of  a  fourth  circle  such 
that  the  four  shall  have  a  common  radical  axis. 

Find  the  radical  centre  of  the  three  circles 

12.  x2  +  2/'  — 4x  +  82/  — 5  =  0,  x^  +  2/' —  8x  — 10^ -f  25  =  0, 

^'  +  2/'  +  8a;  +  111/  —  10  =  0. 

13.  x2  +  2/2  +  6x-82/  +  9  =  0,  x^  +  2/'^  +  8x -f- 22/ +  9  =  0, 

2(^2  +  2/')  -  5(3x  +  2/)  +  18  -  0. 


134.]  THE  CIBCLE.  211 

14.  What  is  the  analytic  condition  that  the  origin  shall  be  the  radical 
centre  of  three  given  circles  ? 

15.  Find  the  equation  of  the  circle  through  the  origin  and  the  points  of 
intersection  of  the  circles 

x'^-i-y^  —  bx  —  ly  +  6  =  0  and  x^ -\- y^ -\- ix -{- 6y  —  12  =  0. 

What  is  the  ratio  of  the  tangents  drawn  from  any  point  on  it  to  the  two 
given  circles  ? 

16.  Find  the  equation  of  the  circle  which  touches  the  line  iy  =  3x  and 
passes  through  the  common  points  of 

a;2  -f  2/2  =  9  and  x''  -\-  y-  -\-  X  -\-2y  =  14. 

17.  What  is  the  ratio  of  the  tangents  drawn  from  any  point  on  the  third 
circle  in  Ex.  11  to  the  other  two  circles? 

18.  Find  the  equations  of  the  straight  lines  which  touch  both  of  the 
circles  x^  +  2/^  =  4  and  (x  —  4)^  -j-  y^  =  1. 

Ans.    3x±v^ly  =  S  and  x±VY5y  =  S, 

19.  Find  the  equations  of  the  common  tangents  to  the  circles 

x'--\-y'-\-6y-\-5^0    and    x'-\-y^  —  12y -^20  =  0. 

20.  If  the  length  of  the  tangent  from  the  point  {x\  y^)  to  the  circle 
x^-\-y-  =  ^  is  twice  the  length  of  the  tangent  from  the  same  point  to 
«'  +  y'  -i-Sx  —  Qy  =  0,  show  that 

a./2  _^  2//2_|.  43./_32,/_|_3  ^  0. 

21.  If  the  tangenc  from  Pto  the  circle  x^ -\- y^ -\- 3y  =  0  is  four  times  as 
long  as  the  tangent  from  P  to  the  circle  x^  -j-  2/^  =  9>  show  that  the  locus  of 
Pis 

5(x-^-f-2/2)=2/  +  48. 

22.  The  length  of  a  tangent  drawn  from  a  point  P  to  the  circle 
x^-\-y^-^ix  —  6i/  +  4  =  0  is  three  times  the  length  of  the  tangent  from  P 
to  the  circle  x'^ -\- y^  —  6x -\- 2y -\- 6  =  0.    Find  the  locus  of  P. 

23.  Find  the  locus  of  a  point  whose  distance  from  the  origin  is  equal  to 
the  length  of  the  tangent  drawn  from  it  to  the  circle 

x^-]-y^  —  Sx  —  iy-\-i=0. 

24.  Find  the  locus  of  a  point  P  whose  distance  from  a  iBxed  point  is  in 
a  constant  ratio  to  the  tangent  drawn  from  P  to  a  given  circle.  Under 
what  condition  is  the  locus  a  straight  line  ? 

25.  Show  that  the  polar  of  any  point  on  the  circle 

x'-\-y^  —  2ax  —  3a-'  =  0, 

with  respect  to  the  circle  x^  -\-y^-\-  2ax  —  3a^  =  0,  will  touch  the  parabola 
y^  +  4ax  =  0. 


212  the  circle.  [135. 

Systems  of  Circles. 
135.     Coaxial  circles. 
Let  S  =  0  and  S'=0  represent  two  circles. 

Then  S-^XS'=0,  (1) 

for  any  value  of  A,  represents  a  circle  through  the  points  of  inter- 
section of  /S  =  0  and  aS"=  0  (§  133).  Moreover,  by  assigning  to  A 
all  possible  values,  it  can  be  made  to  represent  all  the  circles 
which  pass  through  these  points;  that  is,  it  represents  the  entire 
system  of  circles  such  that  the  radical  axis  (§  133)  of  any  pair  of 
circles  of  the  system  is  the  same  as  the  radical  axis  of  the  circles 

Circles  which  have  a  common  radical  axis  are  called  a  System 
of  Coaxial  Circles. 

If  /S  =^  0  is  the  equation  of  a  circle,  and  L  =  0  is  the  equation 
of  a  straight  line,  then  will 

S-\-^L=0  V  (2) 

be  the  equation  of  a  system  of  coaxial  circles  whose  radical  axis 
is  the  line  L  =  0. 

Let  S^=x'^-{-if-{-c,     L=x,     and     A=2gr. 

Then  equation  (2)  becomes 

x'Jt-f-}-2gx-hc  =  0,  (3) 

which  represents  a  system  of  coaxial  circles  whose  centres  are 
( — g,  0),  and  whose  radical  axis  is  the  ^/-axis. 

For,  whatever  value  g  may  have,  the  circle  (3)  cuts  the  ?/-axis 
in  the  same  two  points  (0,  ±  l^ —  c). 

Therefore,  the  radical  axis  meets  these  circles  in  real  points  if 
c  is  negative,  in  imaginary  points  if  c  is  positive. 

Equation  (3)  may  be  written 

(.x  +  9y+rf  =  f~o.  (4) 

Hence,  if  g  is  taken  equal  to  ±  ]/c,  the  circle  will  reduce  to 
one  of  the  points  (dz  Vc,  0). 

These  two  points  are  called  the  Limiting  Points  of  the  sys- 
tem of  coaxial  circles.     When  the  circles  intersect  in  imaginary 


136.]  THE   CIRCLE.  213 

points,  that  is,  when  c  is  positive,  the  limiting  points  are  real; 
and  conversely,  when  the  circles  intersect  in  real  points,  that  is, 
when  c  is  negative,  the  limiting  points  are  imaginary. 

1 36.    To  find  the  condition  that  two  circles  shall  intersect  orthogonally. 

The  two  circles 

0^2+  2/'+  '^9^  H-  2/2/  +  c  =  0,  (1) 

a^-\-  f+  2g'x  +  2f'y  +  c'=: 0,  (2) 

will  intersect  at  right  angles  if  the  distance  between  their  centres 


is  equal  to  the  sum  of  the  squares  of  their  radii ;  i.  e.  if  [§§  7  and 
129] 

or  2gg'+2fr—c-cf=0.  (.3) 

Ex.  1.    Show  that  the  circles 
^'  +  2/'  —  2ax  —  2hy  —  2a6  =  0  and  x'-\-y''-\-  2bx  +  2ay  —  2ab  =  0 
cut  one  another  at  right  angles. 

Ex.  2.    Show  that  the  two  circles 

a;2  +  2/'  —  6x  +  22/  —  22  =  0  and  x""  -\-  y'  -{-  ix  —  6y  -\-  4:  =  0 
intersect  orthogonally.    Find  the  equation  of  a  third  circle  which  shall  cut 
each  of  the  given  circles  at  right  angles. 

Ex,  3.    Find  the  equation  of  a  circle  which  passes  through  the  origin 
and  cuts  orthogonally  each  of  the  circles 

aj'  +  2/'  — 6a;  +  8  =  0  and  x^ -}~y^  —  2x  —  2y  =  1. 

Ex.  4.    Find  the  equation  of  the  circle  which  intersects  orthogonally  each 
of  the  circles 

a;2  4- y2  _  6x  —  62/ + 14  ==  0, 

ic'  +  2/'+6x  — 4t/  +  4  =  0, 
x^+y^—2x  +  6y  =  6. 

Ans.    n(x^  +  2/0  —  10(2a;  +  by)  =  28. 


214  'THE   CIRCLE.  [137. 

137.     1}  a  circle  cuts  orthogonally  each  of  the  circles 

x^^f^2gx+2fy+c  =  0,  (1) 

and  x'-{-f-^  2g'x  +  2fy  +  c'=  0,  (2) 

it  will  cut  orthogo7ially  every  circle  of  the  coaxial  system 

x'-\-f-\-  2gx  +  2/^  +  c  +  X(x'  +  f  -\-2g'x  +  2fy  +  c')  =  0, 

or         .^  +  ,»  +  2(^3±if:).+2(4±¥),  +  ^'=0.      (3) 

Let  the  circle 

x'-i-y'-{-  2Gx  +  2Fy  +  0  =  0  (4) 

cut  both  (1)  and  (2)  orthogonally;  then  [(3),  §  136] 

2gG-^2fF-c-C  =  0,  (5) 

and  2g'G  +  2fF—c'—C  =  0.  (6) 

If  we  multiply  (6)  by  A,  add  the  result  to  (5),  and  then  divide 
by  1  +  A,  we  obtain 

Therefore  (3)  and  (4)  intersect  orthogonally  [§  136,  (3)]. 

CoR.  I.  If  a  circle  cuts  orthogonally  a  system  of  coaxial  circles,  its 
centre  lies  on  the  radical  axis  of  the  system. 

The  radical  axis  of  the  system  of  coaxial  circles  represented 
by  (3)  is  the  line  [(6),  §  133] 

2(g-g')x-h2(f-f)y-^c-c'^=0.  (8) 

Subtracting  (6)  from  (5)  gives 

2(g-g')G-\-2(f-f)F-c-\-c'=^0,  (9) 

which  shows  that  the  point  ( — G,  — F),  the  centre  of  (4),  lies 
on  the  line  (8). 

CoR.  II.  If  the  circles  Si  =  0  and  ^Sg  =  0  cut  the  circles  /Sg  =  0  and 
S^=^0  orthogonally,  then  all  the  circles  of  the  coaxial  system  Si  -\-  aS2  =  0 
will  cut  orthogonally  all  the  circles  of  the  coaxial  system  S2  +  ^A  =  0, 
and  the  locfus  of  the  centres  of  each  system  is  the  radical  axis  of  the  other 
system. 


137.] 


THE   CIRCLE. 


215 


The  equations  of  two  systems  of  coaxial  and  orthogonal  circles 
referred  to  their  radical  axes  as  axes  of  coordinates  (§  133,  Ex.  1) 
may  be  written 

x'  +  y'  +  2gx  —  c  =  0,  (10) 

and  *     x'^y~  +  2Sij-\-c  =  0,  (11) 

where  c  is  constant,  and  g  and /are  arbitrary. 

For  these  equations  represent  two  coaxial  systems  [§  135,  (3)] , 
and  satisfy  the  condition  (3)  of  §  136  for  all  values  of  g  and/. 

The  system  of  circles  represented  by  (10)  intersect  in  the 
points  (0,  ±  \/c)j  which  are  the  limiting  points  (§135)  of  the 
system  represented  by  (11). 


The  heavy  circle  in  the  figure  is  the  smallest  one  of  the  system 
represented  by  (10)  and  corresponds  to  the  value  g=0.  If  this 
circle  is  considered  as  the  boundary  of  the  map  of  a  hemisphere, 
the  other  circles  of  the  same  system  are  the  meridians,  while  the 
circles  of  the  other  system  are  the  parallels  of  latitude  on  this 
map,  thrown  on  the  usual  stereographic  projection. 

Hence, /and  g  may  be  said  to  represent  the  latitude  and  longi- 
tude of  any  point  on  the  map.     (See  §  73. ) 


216  THE   CIRCLE.  [137. 

Examples  on  Chapter  IX. 

1.     Show  that  the  two  circles 

x2  +  2/2  +  2sra;  +  2/V  +  c  =  0  and  x'  +  y''-\-2g'x  +  2f'y-^&=0 
are  tangent  to  each  other  if 


2.  Find  the  equation  of  the  circle  whose  diameter  is  the  common  chord 
of  the  circles 

x'  +  2/'  +  2x  — %  — 4  =  0  and  a;2  +  2/2_6a;  +  42/  +  4  =  0. 

3.  Find  the  equation  of  the  straight  lines  joining  the  origin  to  the  points 
of  intersection  of  the  line  2a;  +  3?/  =  7  and  the  circle  x^-^-y"^  —  4a;  +  2?/  =  0, 
and  show  that  these  lines  are  at  right  angles. 

4.  Find  the  equation  of  the  straight  lines  joining  the  origin  to  the  points 
in  which  the  line  y  =  mx  -\-  c  intersects  the  circle  x'^  +  2/^  =  2(ax  -f  by). 
Hence  find  the  condition  that  these  points  may  subtend  a  right  angle  at 
the  origin.    Also  find  the  condition  that  the  line  may  touch  the  circle. 

5.  A  point  moves  so  that  the  square  of  its  distance  from  a  fixed  point 
varies  as  its  perpendicular  distance  from  a  fixed  straight  line ;  show  that  it 
describes  a  circle. 

6.  Find  the  locus  of  a  point  which  moves  so  that  the  square  of  its  dis- 
tance from  the  base  of  an  isosceles  triangle  is  equal  to  the  product  of  its 
distances  from  the  other  two  sides. 

7.  A  point  moves  so  that  the  sum  of  the  squares  of  the  perpendiculars 
let  fall  from  it  on  the  sides  of  an  equilateral  triangle  is  constant  (k) ;  prove 
that  its  locus  is  a  circle.  Where  is  its  centre?  For  what  value  of  k  will 
the  circle  touch  the  sides  ?    pass  through  the  vertices  of  the  triangle  ? 

8.  Tangents  are  drawn  from  the  point  (h,  k)  to  the.circle  x"^ -\- y^  =  r"^ ; 
prove  that  the  area  of  the  triangle  formed  by  them  and  their  chord  of  con- 
tact is 

y(/t2_^fc2_y2)f 

h""  +  k' 
9.    Find  the  polar  equation  of  a  circle,  the  initial  line  being  a  tangent* 

10.  Prove  that  the  equations 

p  =  a  cos  (6  —  a)  and  p  =  b  sin  (6  —  a) 
represent  two  circles  which  intersect  orthogonally. 

11.  Find  the  polar  equation  of  a  circle  whose  centre  is  the  point  (a,  b), 
(in  rectangular  coordinates)  and  whose  radius  is  r. 

12.  Find  the  polar  equation  of  the  tangent  to  a  circle  at  a  given  point, 
the  centre  of  the  circle  being  at  the  pole. 


137.]  THE  CIRCLE.  217 

13.  What  curve  is  represented  by  the  equation 

P^  —  pa  cos  20  sec  0  —  2a'  =  0  ? 

14.  Detennine  the  locus  of  the  equation 

P  — a  cos (6^  —  a)-\-bcos(0  —  /?)-f-ccos(^  —  7)4-  .  .  . 

15.  The  axes  being  inclined  at  an  angle  w,  find  the  centre  and  radius  of 
the  circle 

x^  +  2xy  cos  o)-\-y'-  —  2gx  —  2fy  -  0. 

16.  The  axes  being  inclined  at  60°,  find  the  equation  of  the  circle  whose 
centre  is  ( —  3,  —  5)  and  radius  6. 

17.  Find  the  locus  of  a  point  which  moves  so  that  the  square  of  the 
tangent  drawn  from  it  to  the  circle  x''-\-y''  =  r'  is  equal  to  a  times  its  dis- 
tance from  the  line  Ix  -{-  my  -\-  n  =  0. 

18.  Find  the  locus  of  a  point  such  that  tangents  from  it  to  two  given 
circles  are  inversely  as  their  radii. 

19.  Find  the  locus  of  the  vertex  of  a  triangle,  having  given  (1)  its  base 
and  the  sum  of  the  squares  of  its  sides,  (2)  its  base  and  the  sum  of  m  times 
the  square  of  one  side  and  n  times  the  square  of  the  other. 

20.  Show  that  the  equation  of  the  circle  circumscribing  the  triangle 
formed  by  the  lines  x-\-y  =  6^  2x  +  2/  =  4,  and  x-\-2y  =  b  is 

x'  +  2/'- 17a;— 1% +  50  =  0. 

21.  Show  that  the  radical  axis  of  two  circles  bisects  their  four  common 
tangents. 

22.  If  Q  is  one  of  the  limiting  points  of  a  system  of  coaxial  circles, 
show  that  the  polar  of  Q  with  respect  to  any  circle  of  the  system  passes 
through  the  other  limiting  point,  and  is  the  same  for  all  circles  of  the 
system. 

23.  If  Q  is  one  of  the  limiting  points  of  a  system  of  coaxial  circles, 
show  that  a  common  tangent  to  any  two  circles  of  the  system  will  subtend 
a  right  angle  at  Q. 

24.  Tangents  are  drawn  to  a  circle  from  any  point  on  a  given  line; 
prove  that  the  locus  of  the  middle  point  of  the  chord  of  contact  is  another 
circle. 

25.  Find  the  locus  of  the  middle  points  of  chords  of  the  circle 
a;2  -f-2/2  _  ^2  -vvhich  subtend  a  right  angle  at  the  point  (a,  0). 

26.  Prove  that  the  square  of  the  tangent  drawn  from  any  point  on  one 
circle  to  another  circle  is  equal  to  twice  the  product  of  the  distance  be- 
tween the  centres  and  the  perpendicular  distance  of  the  point  from  the 
radical  axis  of  the  two  circles. 


218  THE   CIRCLE.  [137. 

27.  Find  the  equations  of  the  straight  lines  joining  the  origin  to  the 
points  of  intersection  of 

a;2  _|_  2,2  _  4a;  _  22/ =  4  and  x2  +  2/2  +  2x  +  4i/ =  10. 

28.  The  distances  of  two  points  from  the  centre  of  a  circle  are  propor- 
tional to  the  distances  of  each  from  the  polar  of  the  other. 

29.  If  the  circle  x^  +  2/^  +  2srx  +  2fy  +  c  =  0  cuts  the  parabola  y"^  =  4ax 
in  four  points,  the  algebraic  sum  of  the  ordinates  of  those  points  will  be 
zero.    (See  §91.) 

30.  If  the  normals  at  the  three  points  P,  Q,  and  i2  of  a  parabola  meet 
in  a  point,  then  the  circle  through  P,  Q,  and  R  will  pass  through  the  vertex 
of  the  parabola.     (§  128  and  Ex.  29  above.) 

31.  The  distances  from  the  origin  to  the  centres  of  three  circles 
/c2_j_  2/2  —  2ax  =  a^  (where  a  is  constant  and  A  variable)  are  in  geometrical 
progression;  prove  that  the  tangents  drawn  to  them  from  any  point  on 
x^-j-y"^  =  r^  are  also  in  geometrical  progression. 

32.  The  polar  of  Pwith  respect  to  the  circle  x^-\-y^  =r'^  touches  the 
circle  (x  —  a)"^  +  (2/  —  by=ri^;  prove  that  the  locus  of  Pis  the  curve 
given  by  the  equation 

ri\x^  -\- y^)  =  {ax -\- by  —  r'^y. 

33.  A  tangent  is  drawn  to  the  circle  (x  —  a)2-|-2/^  =  &^  and  a  perpen- 
dicular tangent  to  the  circle  (x -j- ay -{- y^  =  c^ ;  find  the  locus  of  their 
point  of  intersection,  and  prove  that  the  bisector  of  the  angle  between 
them  always  touches  one  or  other  of  two  circles. 

34.  Show  that  the  equation  of  the  circle  whose  diameter  is  the  common 
chord  of  the  two  circles 

^.2  _j_  2,2  —  2ax  and  x'^-\-y'^  =  2by 
is  (a''-\-b'')ix'-\-y^)=2ab{bx-\-ay). 

35.  Prove  that  the  length  of  the  common  chord  of  the  two  circles  whose 
equations  are 

(x  —  ay  +  (2/  —  by  =  r^  and  (x  —  by  -\-{y  —  ay  =  r\ 


is  ■  V^r'  —  2{a  —  by. 

Hence  find  the  condition  that  the  two  circles  may  touch. 

36.  The  polar  equation  of  the  circle  on  the  line  joining  the  points  (a,  a) 
and  (6,  /3)  as  diameter  is 

p2  — p[a  cos  {d  —  a)-\-b  cos  (^  — y3)]  +  ab  cos  («  — /3)  =  0. 

37.  The  polars  of  a  point  P  with  respect  to  two  fixed  circles  meet  in  the 
point  Q.  Prove  that  the  circle  on  PQ  as  diameter  passes  through  two  fixed 
points,  and  cuts  both  of  the  given  circles  orthogonally. 


137.]  THt   CIRCLE.  219 

38.  Prove  that  the  two  circles,  which  pass  through  the  two  points  (0,  a) 
and  (0,  —  a)  and  touch  the  line  y  =  mx  -f  &,  will  cut  orthogonally  if 

62=aH2H-mO. 

39.  Find  the  coordinates  of  the  centre  of  the  circle  inscribed  in  the  tri- 
angle the  equations  of  whose  sides  are 

3a;  =  42/,    lx  =  2iy,    and    5a:  — 12i/  =  36. 

40.  O  is  any  point  in  the  plane  and  OPiP.  any  chord  of  a  circle  which 
meets  the  circle  in  Pi  and  P^.  On  this  chord  a  point  Q  is  taken  such  that 
OQ  is  equal  to  (1)  the  arithmetic,  (2)  the  geometric,  and  (3)  the  harmonic 
mean  between  OPi  and  OPz>    In  each  case  find  the  locus  of  Q. 

41.  Find  the  locus  of  the  intersection  of  the  tangent  to  any  circle  and 
the  perpendicular  let  fall  on  this  tangent  from  a  fixed  point  on  the  circle. 

42.  A  point  moves  so  that  the  sum  of  the  squares  of  its  distances  from 
the  sides  of  a  regular  polygon  is  constant.    Show  that  its  locus  is  a  circle. 

43.  Show  that  the  locus  of  the  poles  of  tangents  to  the  parabola  y'^=  iax 
with  respect  to  the  circle  x'^  +  y^=  2ax  is  the  circle 

X^  -j-  y'^  =z  ax. 

44.  A  straight  line  moves  so  that  the  product  of  the  perpendiculars  on 
it  from  two  fixed  points  is  constant.  Prove  that  the  locus  of  the  feet  of 
these  perpendiculars  is  a  circle,  the  same  for  each. 

45.  A  straight  line  moves  so  that  the  sum  of  the  perpetidiculars  on  it 
from  two  fixed  points  is  constant.  Find  the  locus  of  the  point  midway  be- 
tween the  feet  of  these  perpendiculars. 

46.  O  is  a  fixed  point  and  AP  and  BQ  are  two  fixed  parallel  lines;  BOA 
is  perpendicular  to  both  and  POQ  is  a  right  angle.  Prove  that  the  locus  of 
the  foot  of  the  perpendicular  drawn  from  O  on  PQ  is  the  circle  on  AB  as 
diameter. 

47.  Find  the  locus  of  a  point  from  which  two  circles  subtend  th^  same 
angle. 

48.  A,  Bj  Cf  and  D  are  four  points  in  a  straight  line.  Prove  that  the 
locus  of  a  point  P,  such  that  the  angles  APB  and  CPD  are  equal,  is  a  circle. 

49.  If  two  points  A  and  B  are  harmonic  conjugates  with  respect  to  C  and 
£>,  the  circles  on  AB  and  CD  as  diameters  cut  orthogonally. 

50.  In  any  circle  prove  that  the  perpendicular  from  any  point  of  it  on 
the  line  joining  the  points  of  contact  of  two  tangents  is  a  mean  proportional 
between  the  perpendiculars  from  the  point  upon  the  two  tangents. 

51.  From  any  point  on  one  given  circle  tangents  are  drawn  to  another 
given  circle.  Prove  that  the  locus  of  the  middle  point  of  the  chord  of  con- 
tact is  a  third  circle. 


220 


THE    CIRCLE. 


[137. 


52.  If  ABC  is  an  acute-angled  triangle,  P  any  point  in  the  plane,  the 
three  circular  loci, 

PA' =  PB^ -{- PC\     PB^  ^  PC -\- PA\     PC'=PA^-\-PB\ 

will  have  their  radical  centre  at  the  centre  of  the  circle  circumscribing  the 
triangle. 

53.  Prove  that  all  circles  touching  two  fixed  circles  are  orthogonal  to 
one  of  two  other  fixed  circles. 


54.    If  <S>  =  0  and  >Si^=  0  are  the  equations  of  two  circles  whose  radii  are  r 
and  r\  then  the  circles 


will  intersect  at  right  angles. 


55.  If  two  circles  cut  orthogonally,  prove  that  an  indefinite  number  of 
pairs  of  points  can  be  found  on  their  common  diameter  such  that  either 
point  has  the  same  polar  with  respect  to  one  circle  that  the  other  has  with 
respect  to  the  other.  Also  show  that  the  distance  between  such  pairs  of 
points  subtends  a  right  angle  at  one  of  the  points  of  intersection  of  the  two 
circles. 

56.  Show  that  the  equation  of  the  orthogonal  circle  of  three  given  cir- 
cles is 


9u 

/i, 

1 

9u 

/2, 

1 

9s, 

/3, 

1 

ix^-{-y')  + 


Cl,/l,    1 

x  + 

C2,   /2,     1 

C3,  /a,   1 

9i7  ci,  1 

92i    C2,     1 


y  — 


9i, 

fu 

Ci 

92, 

/2, 

Cl 

9s 

/3, 

Cs 

=  0. 


57.  If  AB  is  a  diameter  of  a  given  circle,  the  polar  of  A  with  respect  to 
any  circle  which  cuts  the  given  circle  orthogonally  will  pass  through  B. 

58.  From  the  preceding  example,  considering  the  orthogonal  circle  of 
three  given  circles  as  the  locus  of  a  point  such  that  its  polars  with  respect 
to  the  circles  meet  in  a  point,  prove  that  the  equation  of  the  circle  orthog- 
onal to  each  of  three  circles  is 

^  +  9it    y+fu    PiaJ +/i2/ +  ci    =0- 
^  +  92,    2/ 4-/2,    92X+f2y  +  C2 
^  +  9sy     y+fsf     STaX  4-/32/ +  C3 
Show  tha,t  this  equation  is  the  same  as  that  given  in  Ex.  56. 


CHAPTER  X. 

THE  ELLIPSE  AND  HYPERBOLA. 

1 38.  Standard  equations  of  the  tangent,  polar j  and  normal  to  the 
ellipse  and  hyperbola. 

It  has  been  shown  in  §  120  and  §  121*  that,  if  the  axes  of  the 
curve  are  taken  as  coordinate  axes,  the  equations  of  the  central 
conies  may  be  written  in  the  standard  form 

a'  -  W  '  ^^ 

Then  the  coordinates  of  the  foci  are  (  zb  ae,  0)  ;  the  equations 

of  the  directrices  are  a?  =  dz  — ;  the  length  of  the  latus  rectum  is 

e 


2¥         ,         Va'  +  h' 

— ;  and  e  = 

a  a 

For  equation  (1)  formula  (6),  §111,  gives 

Equation.  (2)  is  the  equation  of  the  polar  (§  113)  of  the  point 
{x',  y')  with  respect  to  the  central  conic  (1),  which  polar  is  a 
tangent  at  the  point  (a?',  y')  when  (a?',  ?/)  is  on  the  conic. 

The  equation  of  the  normal  at  any  point  (a/,  yf)  on  the  conic 

^^^''  2/-y=^(^-^'),         [(2),  §85] 

or  ;—  =  ^     ,'' .  (3) 

x'  y  ^  ^ 

"^  '±b' 

Ex.  1.  Find  the  equations  of  the  central  conies  when  the  origin  is  at 
either  focus;  at  either  vertex;  at  the  point  (h,  fc),  the  coordinate  axes 
being  parallel  to  the  axes  of  the  conic. 

Ex.  2.  What  relation  does  the  line  (3)  have  to  the  conic  when  (x%  y^) 
is  not  on  the  curve  ? 

♦These  sections  should  now  be  carefully  reviewed. 

t  We  shall  use  this  form  of  the  equation,  although  the  simpler  form  ax^  +  62/2  =  1  is 
sometimes  more  convenient.  When  the  double  sign  ±  or  t  is  prefixed  to  h-,  the  upper  sign 
holds  for  the  ellipse  and  the  lower  for  the  hyperbola.  All  results  are  true  for  both  curves 
unless  the  contrary  is  expressly  stated.  Furthermore,  results  for  the  ellipse  include 
those  for  the  circle  as  the  special  case  when  a  =  6. 


222  THE   ELLIPSE   AND    HYPERBOLA.  [139, 

1 39.     To  find  the  equation  of  the  tangent  to  the  conic 

in  terms  of  its  slope  m. 

Assume  the  equation  of  the  tangent  to  be 

y  =  mx^c,  (2) 

where  m  is  known,  and  c  is  to  be  determined  so  that  (1)  and  (2) 
shall  intersect  in  two  coincident  points  (§  78). 
Eliminating  y  between  (1)  and  (2)  gives 

x^       {mx  +  c)'^  _ 
^-  P         ~^' 

or  ir'(aW=t60  +  2a'cma;  +  a'(c=^=P  60=0.  (3) 

The  roots  of  equation  (3)  will  be  equal  if 

aXe"  q=  h')  {aV  ±.  b')  =  a'cV, 

Whence  c' =  aW  ±  b\  (4) 

That  is,  the  points  of  intersection  of  the  straight  line  and  the 
conic  will  coincide  if 

c  =  +  VaW  ±  b\  (5) 

Hence  the  line  whose  equation  is 


y  =mx  i^VaV  ±  W,  (6) 

will  touch  the  conic  (1)  for  all  values  of  m. 

The  double  sign  before  the  radical  in  (6)  shows  that  there  are 
two  tangents  for  every  value  of  m ;  i.  e.  there  are  two  tangents  to 
a  central  conic  parallel  to  any  given  straight  line ;  and  these  two 
parallel  tangents  are  equidistant  from  the  center  of  the  conic. 

Ex.  1.    Derive  equation  (6)  by  the  method  used  in  §  125. 

Ex.  2.  In  a  similar  manner  show  that  the  equation  of  the  normal  to  (1) 
expressed  in  terms  of  its  slope  is 

y  =  rax  —  -^  — . 

^  Va''±  b'm^ 

Ex.  3.  How  many  normals  can  be  drawn  from  a  given  point  to  a  central 
conic  ? 


139.]  THE   ELLIPSE   AND    HYPERBOLA.  223 

EXAMPLES. 

Find  the  eccentricity,  foci,  and  latus  rectum  of  each  of  the  following 
conies : 

I.  a;2  +  23/2=4.  2.    ix'  —  dy'=SQ. 
3.    4x2  +  3/2=8.  4.    3x2  —  2/2=9. 

5.    3(x- 1)2 +  4(3/ +  2)2=1.  6.    3(3/-l)2-4(x  + 1)2=1. 

Find  the  equation  of  an  ellipse  referred  to  its  axes " 

7.  if  the  latus  rectum  is' 6  and  the  eccentricity  i. 

8.  if  the  latus  rectum  is  4  and  the  minor  axis  is  equal  to  the  distance 
between  the  foci. 

9.  Find  the  equation  of  the  hyperbola  whose  foci  are  the  points  (rb  4, 0) 
and  whose  eccentricity  is  |/2. 

10.  Find  the  eccentricity  and  the  equation  of  the  ellipse,  if  the  latus 
rectum  is  equal  to  half  the  minor  axis. 

II.  Find  the  equation  of  the  hyperbola  with  eccentricity  2  which  passes 
through  (—4,6). 

12.  Find  the  equati-on  of  the  ellipse  passing  through  the  points  ( — 2,  2) 
and  (3, — 1);  also  the  equation  of  the  hyperbola  through  (1, — 3)  and 
(2,4). 

Through  how  many  points  can  a  central  conic  be  made  to  pass  if  its  axes 
are  given?    Why? 

13.  Find  the  eccentricity  and  the  equation  of  a  central  conic  if  the  foci 
lie  midway  between  the  centre  and  the  vertices;  if  the  vertices  lie  midway 
between  the  centre  and  the  foci. 

14.  Show  that  the  tangents  at  the  ends  of  either  axis  of  a  central  conic 
are  parallel  to  the  other  axis ;  and  also  that  tangents  at  the  ends  of  any 
chord  through  the  centre  are  parallel. 

15.  Find  the  equations  of  the  tangents  and  normals  at  the  ends  of  the 
latera  recta.    Where  do  they  meet  the  x-axis?  One  Ans.    3/  +  ex  =  a. 

16.  Show  that  the  line  3/  =  2x  —  y^|  touches  the  conic 

3x2  —  63/2=1. 

17.  Find  the  equations  of  the  tangents  to  the  ellipse  x2  +  43/2  =  16  which 
make  angles  of  45°  and  60°  with  the  x-axis. 

18.  Show  that  the  directrix  is  the  polar  of  the  focus. 

19.  If  the  slope  of  a  moving  line  remains  constant,  the  locus  of  its  pole 
with  respect  to  a  central  conic  is  a  straight  line  through  the  centre  of  the 
conic. 


224  THE   ELLIPSE   AND    HYPERBOLA.  [139. 

20.  Show  that  the  minor  axis  is  a  mean  proportional  between  the  major 
axis  and  the  latus  rectum. 

21.  Show  that  the  ellipse  is  concave  towards  both  axes,  while  the  hyper- 
bola is  concave  only  towards  its  transverse  axis 

22.  Show  that  the  line  Ix  +  my  —  n  will  touch 

^,±|r,  =  1    if    aH^  ±  b'm'' =  n\ 
a^      b^ 

The  line  x  cos  a-^y  sin  a  =p  will  touch  the  same  curves  if 

a^  cos^  a  ±b^  sin^  a  =p'^. 

23.  Show  that  the  point  (cci,  2/1)  is  inside,  on,  or  outside  the  ellipse 


x^,y^ 
a"  "^  h' 
according  as 


4--  =1 

2     I     K2  -^ 


2  1,  2 


^+^-l<,   =,  or>0. 


24.    Show  that  the  point  (%,  y-C)  is  inside,  on,  or  outside  the  hyperbola 
according  as 


a?      b' 


^-f -1>,   ==,  or<0. 

25.  Are  the  points  (3,  |),  ( —  |,  —  v''^),  (5,  —  2)  inside  or  outside  of  the 
curves  bx"^  ±  dy"^  =  50  ? 

26.  Find  the  equations  and  the  coordinates  of  the  points  of  contact  of 
tangents  to  b'^x^  ±  a'^y^  =  aW  which  make  equal  intercepts  on  the  axes. 

27.  If  the  normal  at  the  end  of  the  latus  rectum  of  an  ellipse  passes 
through  the  extremity  of  the  minor  axis,  show  that  the  eccentricity  is  given 
by  the  equation  e*  +  6^  =  l-  Find  the  corresponding  equation  for  the 
hyperbola  and  interpret  the  result. 

28.  If  any  ordinate  MP  of  a  central  conic  is  produced  to  meet  the  tan- 
gent 9,t  the  end  of  the  latus  rectum  through  the  focus  F  in  Q,  show  that 
FP  =  MQ. 

29.  Find  the  product  of  the  segments  into  which  a  focal  chord  of  a  cen- 
tral conic  is  divided  by  the  focus. 

30.  Two  tangents  can  be  drawn  to  a  central  conic  from  any  point,  which 
will  be  real,  coincident,  or  imaginary  according  as  the  point  is  outside,  on, 
or  inside  the  conic.    Thus  determine  which  is  the  inside  of  a  hyperbola. 


140.] 


THE   ELLIPSE   AND    HYPERBOLA. 


226 


140.     Conjugate  Hyperbolas. 

The    two    hyperbolas    whose 
equations  ai  e 


and 


or 


a? 


,»       52  -       h 


Z.2  „2  —   ^1 


(1) 


!►      (2) 


are  so  related  that  the  transverse 
axis  of  the  one  is  the  conjugate 
axis  of  the  other. 

The  two  hyperbolas  are  then 
said  to  be  be  conjugate  to  one 
another. 


The  eccentricity  of  the  Conjugate  Hyperbola^  is  e^  = 


t/6^+  a' 


the  coordinates  of  its  foci  are  (0,  db6ei);  the  equations  of  its 
directrices  are  «  =  ±  — ;  and  its  latus  rectum  is  -^. 

When  a  =  6   equations  (1)  and  (2)  become,  respectively, 


and 


x'-,f=a^^ 
y'~x'=a\j 


(3) 


Hence  if  a  hyperbola  is  equilateral  or  rectangular  [§  121,  (15)], 
its  conjugate  is  also  rectangular. 

Two  conjugate  hyperbolas  are  not,  in  general,  similar  (§  116), 
i.  e.  of  the  same  shape,  but  two  conjugate  rectangular  hyperbolas 
are  similar  and  equal. 

*The  hyperbola  (2)  is  usually  called  the  Conjugate  Hyperbola,  while  (1)  is  called  the 
Original,  or  Primary  Hyperbola.  It  is  to  be  noticed  that  the  equation  of  the  conjugate 
hyperbola  is  found  by  changing  the  sign  of  one  member  of  the  equation  of  the  primary 
hyperbola.    Likewise  thfe  equation  of  the  conjugate  ellipse  is  found  to  be 

?-"  +  ?^  =  _  1 
o2  ^  62 

Hence  the  conjugate  of  an  ellipse  is  imaginary. 

The  student  should  compare  the  results  given  above  with  the  conjugate  properties 
discussed  in  §  116.  ' 

16 


226  THE    ELLIPSE    AND    HYPERBOLA.  [141. 

141.     To  find  the  locus  of  the  point  of  intersection  of  two  perpen- 
dicular tangents  to  the  conic 

The  equation  of  any  tangent  to  (1)  may  be  written  (§  139) 


y  zizmx  -\-  Vd'm'  ±  b\  (2) 

If  this  line  (2)  passes  through  {oc^,  ?/i),  we  shall  have 
2/i  =  mxi  +  V^aV  ±  b'\ 
which  when  rationalized  becomes 

(x,'—a')m'  —  2x,y,m  -{-(y,'  +  b')  =  0.  (3) 

This  equation  is  a  quadratic  in  m  whose  two  roots  are  the  slopes 
of  the  two  tangents  which  pass  through  the  point  (xi,  yi),  whose 
locus  is  required. 

Let  m,  and  wig  be  the  two  roots  of  (3)  ;  then  (§91) 

y,-  ^  b" 

x.'  —  a" 

The  two  tangents  will  be  at  right  angles  if  m^nii  =  —  1  (§  48)  ; 
i.  e.  if 


^.--1, 


or  x,'^y,'=d'±b\  (4) 

The  required  locus  is,  therefore,  the  circle 

x'  +  2/'  =  «'  ±  ^\  (5) 

which  is  called  the  Director  Circle  of  the  conic. 

CoR.  1.     If  a  <ib,  the  director  circle  of  a  hyperbola  is  imaginary. 

Hence  one  of  the  director  circles  of  two  conjugate  hyperbolas  is  always 

imaginary.  ,.. 

x^        iP" 
CoR.   II.      The   director   circle   of  the   ellipse  — y  -|-  -^  =  1  parses 

through  the  foci  of  the  hyperbolas  — ^ ^  =  =t  1?  ^^^  '^^^^  versa. 

a  0 

What  does  this  mean  when  a  =  6? 


142.] 


THE   ELLIPSE   AND    HYPERBOLA, 


227 


142.     Auxiliary  Circle,  and  Eccentric  Angle. 

I.     The  circle  described  on  the  major  axis  of  an  ellipse  as 
diameter  is  called  the  Auxiliary  Circle. 


If  the  equation  of  the  ellipse  is 


a'  "^  b' 


(1) 


the  equation  of  the  auxiliary  circle  will  be 

x'^y'  =  a\  (2) 

If  the  ordinate  NP  of  any  point  P  on  the  ellipse  is  produced 
to  meet  the  auxiliary  circle  in  Q,  then  P  and  Q  are  called  Corre- 
sponding Points. 

Let  P(aJi,  2/i)  and  Q{xi,  y^)  be  any  two  corresponding  points; 
then,  since  these  points  are  on  (1)  and  (2),  respectively, 

'     —  (3) 


^1  =  -  Var — x\ 


and 


2/: 


a' — x^ 


(4) 
(5) 


That  is,  the  ordinates  of  corresponding  points  are  in  a  constant 
ratio. 


Ex.    Show  that  the  area  of  the  ellipse  is  irah. 


228 


THE   ELLIPSE  AND   HYPERBOLA. 


[142. 


The  angle  XOQ  is  called  the  Eccentric  Angle  of  the  point  P. 
It  will  be  denoted  by  (p. 

Then  the  coordinates  of  the  point  Q  are 

aci  =  a  cos  <p,     11'^  =  d  sin  c. 

Since  yi  =  —  y.^  =  b  sin  <p,  the  coordinates  of  P  aro 
d 

Xi=  a  cos  ^,     yi=  b  sin  <p.  (6) 

II.     The  circle  described  on  the  transverse  axis  of  a  hyperbola 
as  diameter  may  be  called  the  Auxiliary  Circle  *  of  the  hyperbola. 


Let  P(^x,  y)  be  any  point  on  the  hyperbola  and  NP  its  ordi- 
nate. Draw  NQ  tangent  to  the  auxiliary  circle  at  Q,  so  that  P 
and  Q  are  on  the  same  side  of  the  transverse  axis  when  P  is  on 
the  right  branch,  and  on  opposite  sides  when  P  is  on  the  left 
l)ranch  of  the  curve.  Then,  as  P  describes  the  complete  hyper- 
bola  in  the  direction  indicated  by  the  arrows,  Q  will  move  con- 
secutively around  the  circle  in  the  direction  indicated.  Thus,  for 
every  position  of  P  on  the  hyperbola,  there  is  one  and  only  one 
<;orresponding  position  of  Q  on  the  circle. 

Hence  P  and  Q  may  be  called  Corresponding  Points,  and  the 
angle  XOQ=  <p  may  be  called  the  Eccentric  Angle  *  of  the  point  P. 

♦The  terms  "Auxiliary  Circle"  and  "  Eccentric  Angle  "  are  not  generally  used  with 
reference  to  the  hyperbola,  but  are  here  employed  in  order  to  express  the  coordinates  of 
any  point  on  the  curve  in  terms  of  a  single  variable  9. 


143.]  THE   ELLIPSE   AND   HYPERBOLA,  229 

Let  the  equation  of  the  hyperbola  be 

Then  ON  =  x  =  a  sec  <p,  (8) 

which  substituted  in  (7)  gives 

y  =  h  tan  ^p,  (9) 

That  is,  P  is  the  point  (a  sec  ^,  6  tan  ^). 

Similarly,  ^  -\-  if  —  h^  is  the  auxiliary  circle  of  the  conjugate 
hyperbola,  and  (a  tan  v',  h  sec  ^)  is  any  point  on  the  curve  if  <p 
is  measured  clocJcwise  from  the  positive  end  of  the  y-Sbxia;  if  ^  is 
measured  from  the  a:-axis  the  point  is  (a  cot  <p,  h  esc  s^). 

1 43.  To  find  the  equation  of  the  straight  line  joining  two  points  on 
a  conic  whose  eccentric  angles  are  (f  and  <p'. 

If  the  conic  is  an  ellipse,  the  points  are  (§  142) 

(a  cos  (fj  h  sin  (p)     and     (a  cos  ^',  6  sin  <p'). 

The  equation  of  the  line  through  these  points  is  [(3),  §  47] 

X  —  a  cos  ^       _        y  —  6  sin  ^ 
a  cos  (f  —  a  cos  <p'  ~  h  sin  (p  —  6  sin  (p'' 

Since  cos^  —  cos  <p'=  — 2  sin^(^  +  v')  sin  ^{<p  —  <p') 

and  sin  <p  —  sin  ^'=  2  cos  ^{<p  -\-  (p')  sin  \{(p  —  <p')j 

equation  (1)  reduces  to 

^_cos,.  _      f-^^°^  (2) 


—  2  sin  ^(v  +  /)      2  cos  ^{<p  +  /) 

.-.     |cosK^  +  /)4-|sinKs^  +  S^')=cosK^  — ^0,        (3) 

which  is  the  required  equation. 

In  like  manner  the  equation  of  the  line  joining  the  points 
(a  sec  ^,  h  tan  <p)  and  (a  sec  ^',  h  tan  <p')  on  the  hyperbola  can 
be  shown  to  be 

^  cos  \{<p  —  v')  —  I  sin  ^{<p  +  ^')  =  cos  \{<p  -h  ^')-       (4) 


230  THE   ELLIPSE   AND    HYPERBOLA.  [144. 

To  find  the  equation  of  the  tangent  at  the  point  95,  we  put 
^'  =  ^  in  equations  (3)  and  (4),  and  we  obtain  for  the  ellipse 

X  V 

-  cos  ^  -\-f  sin  <p  =  1,  (5) 

Cb  0 

and  for  the  hyperbola 

or  11 

—  sec  <p  —  J-  tan  <p  =  1.  (6) 

From  equation  (3)  we  see  that  if  the  sum  of  the  eccentric  angles 
of  two  points  on  an  ellipse  is  constant  and  equal  to  2a,  the  equa- 
tion of  the  line  joining  them  is 

—  cos  a -f  ^  sin  a  =  cos  "K^  —  ^').  (7) 

Hence  the  chord  is  always  parallel  to  the  tangent 

—  cos  a -|- |- sin  a  =  1.  (8) 

Conversely,  in  a  system  of  parallel  chords  of  an  ellipse,  the  sum 
of  the  eccentric  angles  of  the  extremities  of  any  chord  is  constant. 

Similarly  from  equation  (4)  we  see  that  if  the  sum  of  the  ec- 
centric angles  of  two  points  on  a  hyperbola  is  constant  and  equal 
to  2a,  the  equation  of  the  chord  through  these  points  is 

fly  1/ 

—  cos^(v  —  <p')  —  ^  sin  a  =  cos  a,  (9) 

and  therefore  the  chord,  and  the  tangent  at  the  point  a,  viz. , 

x      11 

r-  sin^a  =  cos  a,  (10) 

ah 

always  meet  the  2/-axis  in  the  same  fixed  point. 

1 44.  To  find  the  equation  of  the  normal  at  any  point  in  terms  of  the 
eccentric  angle  of  the  point. 

Let  (a  cos  ^,  6  sin  v)  (§  142)  be  any  point  on  the  ellipse;  then 

the  slope  of  the  tangent  at  the  point  s^  is -. — -,    [§  143,(5).] 


144.]  THE    ELLIPSE   AND    HYPERBOLA.  231       - 

Hence  the  equation  of  the  normal  at  ^  is  [(2),  §  85] 

-    .  a  sin  ^ 

y  —  6  sin  ^  = -^ (x  —  a  cos  <p),  (1) 

^  h  cos  (p  ^  ^   ^ 

ax  by  o      ,9  ,^x 

or  r^—  =  a?—  ¥.  (2) 

cos  (p       sin  <p  ^ 

Similarly  we  find  the  equation  of  the  normal  to  the  hyperbola 
at- the  point  (a  sec  ^,  h  tan  ^)  to  be 

sec  ff      tan  <p  ^  ^ 

EXAMPLES. 

i.  Show  that  the  equation  of  the  locus  of  the  foot  of  the  perpendicular 
from  the  centre  of  a  conic  on  a  tangent  is  p^  =  a^  cos'^  d±b  sin-  d.  (See 
Ex.  22,  p.  224.) 

2.  An  ellipse  slides  between  two  perpendicular  lines;  show  that  the 
locus  of  the  centre  is  a  circle.     (§  141.) 

3.  Show  that,  for  all  values  of  6,  tangents  to  the  ellipse    ,^  -f  ?i<  =  1  at 

points  having  the  same  abscissa  meet  the  x-axis  in  the  same  point.    Hence 
show  how  a  tangent  can  be  drawn  to  an  ellipse  from  any  point  on  the  x-axis. 

4.  Two  tangents  are  drawn  to  a  conic  from  any  point  on  the  auxiliary 
circle ;  prove  that  the  sum  of  the  squares  of  the  chords  which  the  auxiliary 
circle  intercepts  on  them  is  equal  to  the  square  of  the  line  joining  the  foci. 
(See  (9),  §  145.) 

5.  If  the  points  Q  and  Q^  are  taken  on  the  minor  axis  of  a  conic  such 
that  QO  =  0Q'=  OFf  where  O  is  the  centre  and  Fa  focus,  show  that  the 
sum  of  the  squares  of  the  perpendiculars  from  Q  and  Q^  on  any  tangent  to 
the  conic  is  constant. 

6.  If  p  is  the  lejjgth  of  the  perpendicular  from  the  centre  on  the  chord 

joining  the  extremities  of  two  perpendicular  diameters  of  an  ellipse,  show 

that 

_        ah 

7.  A  line  is  drawn  through  the  centre  of  a  conic  parallel  to  the  focal     | 
radius  of  a  point  P  and  meeting  the  tangent  at  Pin  Q.    Find  the  locus  of  Q-J 

From  one  focus  of  an  ellipse  a  perpendicular  is  drawn  to  any  tangent 
and  produced  to  an  equal  distance  on  the  other  side.  Show  that  its  ter- 
minus Q  is  in  the  straight  line  through  the  other  focus  and  the  point  of 
tangency.    Also  find  the  locus  of  Q. 


232  THE   ELLIPSE   AND    HYPERBOLA.  [144. 

9.  Show  that  the  locus  of  the  point  of  intersection  of  tangents  to  an 
ellipse  at  two  points  whose  eccentric  angles  differ  by  a  constant  is  an  ellipse. 

[If  the  tangents  at  ^  +  "  ^^^  ^  —  "  meet  at  {x%  y^),  then  —  =  cos  (p  sec  a, 

^  =  sin  <&  sec  a.    Eliminate  <p  for  the  locus.] 
b 

What  is  the  corresponding  theorem  for  the  hyperbola  ? 

10.  The  point  P(—  3,  —  1)  is  on  the  ellipse  x^  -|-3?/  =  12;  find  the  cor- 
responding point  on  the  auxiliary  circle,  and  also  find  the  eccentric  angle 
of  P. 

11.  The  polar  of  a  point  P  with  respect  to  an  ellipse  cuts  the  minor  axis 
in  A ;  and  the  perpendicular  from  P  to  its  polar  cuts  the  polar  in  B  and  the 
minor  axis  in  C.  Show  that  the  circle  through  A,  B,  and  C  will  pass 
through  the  foci. 

[Prove  AO'OC  =  F^O  •  OF,  where  O  is  the  centre.  J 

12.  Prove  that  the  circle  on  any  focal  radius  as  diameter  touches  the 
auxiliary  circle. 

13.  Prove  that  the  line  Ix  +  my  -f-  ^  ==  0  is  normal  to 

„2  -^  52     ^»         12  -r  ^2  ^2 

[Compare  Ix -\- my -\- n  =  0  with  -^  —  -.^  =  a^  —  bK    (See  §  63. )] 

14.  Prove  that  a  circle  can  be  drawn  through  the  foci  of  a  hyperbola 
and  the  points  in  which  any  tangent  meets  the  tangents  at  the  vertices. 

15.  The  perpendicular  from  the  focus  of  an  ellipse  upon  any  tangent 
and  the  line  joining  the  centre  to  the  point  of  contact  meet  on  the  corre- 
sponding directrix. 

16.  If  Q  is  the  point  on  the  auxiliary  circle  corresponding  to  the  point 
P  on  the  ellipse,  the  normals  at  P  and  Q  will  meet  on  the  circle 

x^-^y^  =  ia-\-by. 

17.  Prove  that  the  focal  radius  of  any  point  on  a  central  conic  and  the 
perpendicular  from  the  centre  on  the  tangent  at  that  point  meet  on  a  circle 
whose  centre  is  the  focus  and  whose  radius  is  the  semi-major  axis. 

18.  If  P(x^,  yO  is  a  point  on  an  ellipse,  prove  that  the  angle  between 

b'^ 
the  tangent  at  P  and  the  focal  radius  of  P  is  tan-^ ;. 

19.  If  Q  is  the  point  on  the  auxiliary  circle  corresponding  to  the  point 
P  on  the  ellipse,  show  that  the  perpendicular  distances  of  the  foci  P,  P^ 
from  the  tangent  at  Q  are  equal  to  FP  and  F^P  respectively. 


144.3  THE    ELLIPSE   AND    HYPERBOLA.  233 

20.  If  a  polar  with  respect  to  a  central  conic  touches  the  circle 
x^-^y'^  =  h\  what  is  the  locus  of  the  pole  ? 

21 .  Show  that  the  polar  of  any  point  on  either  of  the  curves 

a'  -  h^ 
with  respect  to  the  other  touches  the  first  curve. 

22.  The  polar  of  any  point  Pon  either  of  the  curves 

a'      b=»      - 
with  respect  to  the  other  touches  the  first  curve  at  the  opposite  extremity 
of  the  diameter  through  P. 

23.  The  polars  of  any  point  with  respect  to  the  two  conies 

a"      52      -  ^ 
are  parallel  and  equidistant  from  the  centre. 

24.  The  product  of  the  focal  radii  of  any  point  on  a  rectangular  hyper- 
bola is  equal  to  the  square  of  the  distance  from  the  centre  to  that  point. 

25.  The  distance  of  any  point  Q  from  the  centre  of  a  rectangular  hyper- 
bola varies  inversely  as  the  perpendicular  from  the  centre  upon  the  polar 
of  ^. 

26.  If  the  normal  at  any  point  P  of  a  rectangular  hyperbola  meets  the 
axes  in  N  and  N\  and  O  is  the  centre,  then  PN=  PN^  =  OP. 

27.  Chords  are  drawn  through  the  end  of  an  axis  of  an  ellipse.  Find 
the  locus  of  their  middle  points. 

28.  Find  the  locus  of  the  pole  of  a  chord  of 

a'  -  b'  ~    * 
which  subtends  a  right  angle  (1)  at  the  centre,  (2)  at  the  vertex,  and 
(3)  at  the  focus  of  the  curve. 

29.  Show  that  the  area  of  a  triangle  inscribed  in  an  ellipse  is 

^a5[sin  (a  —  ^)-\-  sin  (fi  —  y)  +  sin  (7  —  a)], 

where  a,  /5,  y  are  the  eccentric  angles  of  the  vertices. 

Prove  also  that  its  area  is  to  the  area  of  the  triangle  formed  by  the  cor- 
responding points  on  the  auxiliary  circle  as  6:a;  and  hence  its  area  is  a 
maximum  when  the  latter  is  equilateral;  i.  e.  when 

a  <^  l3  =  i3  ^  y  =z  y  ^  a  =  ^7Z. 

30.  If  P  is  a  point  on  the  director  circle  of  an  ellipse,  and  O  the  centre, 
the  product  of  the  distances  of  O  and  P  from  the  polar  of  P  with  respect 
to  the  ellipse  is  constant. 


234  THE   ELLIPSE   AND    HYPERBOLA. 

145.     Geometric  properties  of  the  ellipse  and  hyperbola. 


[145. 


Let  the  tangent  at  P(^',  ;/)  meet  the  axes  in  T'and  T';  let  the 
normal  at  P  meet  the  axes  in  N  and  N')  let  BP  be  the  ordinate 
of  P  and  F^  F'  the  foci  of  the  conic. 

Draw  FG^  F'G',  and  OiT  perpendicular  to  the  tangent  PT. 


Then  0T^-„ 

x' 


0T'  = 


y 


Subtangent  =^  RT  ^ 
ON^  e-x',         ON' 


If- 


[(2),  §  138.] 


[(3),  §138.] 


.•.     Subnormal  =  NR  =  {I  —  e')x'. 
OK-NP  =FG-F'0'=±  b\ 
ON-  OT=a-e-  =  OFK 

PN-PN'=  FP-F'P=±ia'—e'x").     (§§120-1.) 
F'O  and  FG'  bisect  PN 
The  locus  of  the  points  O  and  G'  is  the  circle 

x'+  if=  a\      [Use  (6),  §  139.] 
F'N  _F'0+  ON  _  ae  +  eV  _  a  +  ex' 
NF  "  OF—  ON  "  ae  —  eV  ~  u  —  ex' 


(1) 
(2) 

(3) 

(4) 
(5) 
(6) 
(7) 
(8) 

(9) 


OF—  ON 

ae  —  e'^x' 

F'N 

F'P 

•  •     NF~ 

^FP' 

5§  120-1.)     (10) 


145.] 


THE   ELLIPSE   AND   HYPERBOLA. 


235 


Therefore  the  tangent  and  the  normal  bisect  the  angles  between 
the  focal  radii  FF  and  F'F. 


Hence,  if  an  ellipse  and  a  hyperbola  have  the  same  foci,  the 
tangent  and  the  normal  to  one  of  the  curves  at  any  one  of  their 
four  common  points  are,  respectively,  the  normal  and  the  tangent 
to  the  other.     That  is,  the  two  conies  intersect  orthogonally. 

Conies  having  the  same  foci  are  called  Oonfocal  Conics. 

Ex.  1.  Explain  what  would  happen  if  a  light  were  placed  at  one  focus  of 
an  ellipse ;     a  hyperbola. 

Ex.  2.    What  is  the  Umit  of  ON,  0N%  and  NRaiSx'=a?    as  x'=  0? 

Ex.  3.  Show  that  equations  (1),  (3),  and  OK  ■  NP  =  b'^  are  also  true  when 
P  is  any  point,  TT^  the  polar  of  P,  and  PN  is  perpendicular  to  TT^. 


Ex.  4.    Show  that  the  equation 


y 


a^  +  /  '  5'^  + A 


=  1 


represents  a  system  of  confocal  conics,  where  A  is  the  arbitrary  parameter; 
prove  analytically  that  confocal  conics  intersect  at  right  angles. 


236  THE   ELLIPSE   AND   HYBERBOLA.  [146, 

1 46.  Def.  An  Asymptote  *  is  a  line  which  meets  a  curve 
in  two  points  at  infinity,  but  which  is  itself  not  altogether  at  in- 
finity. 

To  find  the  asymptotes  of  the  hyperbola 

As  in  §  139,  the  abscissas  of  the  points  where  the  line 

y  z=mx  -\-  G  (2) 

meets  the  hyperbola  are  given  by  the  equation 

x\aV  —  b')-\-2a'(mx  +  a\c'  +  b')  =  0.  (3) 

If  the  line  (2)  is  an  asymptote,  both  roots  of  equation  (3) 
must  be  infinite.  Hence  the  coefficients  of  x^  and  x  must  both 
be  zero  (§  98,  III. ).     That  is, 

a^cm  =  0,     and     aV  —  6^  =  0. 

.•.     c  =  0, ,  and    m=±—.  C4> 

a 

Substituting  these  values  in  (2),  we  have  for  the  required 
equations  of  the  asymptotes 

y=±-x,  (5) 

or  expressed  in  one  equation  (§53) 

Therefore  the  hyperbola  has  two  asymptotes,  both  passing 
through  the  centre  and  equally  inclined  to  the  transverse  axis. 

The  equations  of  the  asymptotes  to  a  hyperbola  can  also  be 
found  by  considering  them  the  limiting  positions  of  the  tangent 
as  the  point  of  contact  moves  off  to  infinity. 

The  equation  of  the  tangent  to  (1)  at  (a?',  /)  is 


a'         b'  ~" 


(7) 


*  See  note  under  §  116. 


146.]  THE   ELLIPSE   AND   HYPERBOLA.  237 

Since  the  point  (.r',  y')  is  on  the  conic  (1),  we  have 
Hence  equation  (7)  may  be  written 

-^±y^XZl-\.  (8) 

a-       ah  y         x'      x' 

If  now  the  point  of  contact  (a^',  i/')  moves  off  to  infinity,  so  that 
ip'=  X,  the  limiting  position  of  the  line  (8)  is  given  by  the 
equation 

which  is  the  same  as  equation  (5)  above. 

CoR.  I.  Two  conjugate  hyperbolas  have  the  same  asymptotes^  which 
are  the  diagonals  of  the  rectangle  formed  by  the  tangents  at  their  vertices. 

CoR.  II.  A  straight  line  parallel  to  an  asymptote  will  meet  the  conic 
in  one  point  at  infinity. 

For,  if  c  is  not  zero,  only  one  root  of  (3)  is  infinite. 
Cor.  III.     The  line  y  =mx  will  cut  the  hyperbola  in  real  or  imagi- 
nary points  according  as  m  ^^  or  ^  —.     It  will  meet  either  the  hyperbola 
V  a 

or  its  conjugate  in  real  points  for  all  values  of  m. 

CoR.  IV.     The  asymptotes  of  an  ellipse  are  imaginary. 
For,  if  we  change  the  sign  of  b^,  the  values  of  m  for  infinite 
roots  in  (3)  become  imaginary. 

It  is  to  be  noticed  that  the  equations  of  two  conjugate  hyper- 
bolas and  the  equation  of  their  common  asymptotes,  viz. , 


/^2  ^.2  ^2  ^.2 

-o  —  o  =^  ±  1     and     -,  —  %  =  0, 
a^       b^  a^       b- 


differ  only  in  their  constant  terms.  Moreover,  this  must  always 
be  true ;  for  any  transformation  of  coordinates  will  affect  the  first 
members  of  these  equations  in  precisely  the  same  way.  Hence 
the  new  equations  will  differ  only  in  their  constant  terms  (not 
usually  by  unity)  ;  and  the  value  of  the  constant  in  the  equation 
of  the  asymptotes  will  be  equal  to  half  the  sum  of  the  constants 
in  the  equations  of  the  two  hyperbolas.     {Cf.  §  117.) 


238  THE    ELLIPSE    AND    HYPERBOLA.  [147. 

147.     Similar  and  Coaxial  Conies. 

Since  ay^KebJid  bi/Kare  the  semi-axes  of  the  ellipse 


a'  "^  b 
its  eccentricity  is  given  by  the  equation 


J.--E',  (1) 


^^Va^K^^V^^  (§138.) 

That  is,  the  eccentricity  of  (1)  is  the  same  as  the  eccentricity 
of  the  ellipse  represented  by  the  standard  equatiorc 

f:+i^=i.  (2) 

Therefore  the  two  ellipses  (1)  and  (2)  are  similar  (§  116) 
whatever  value  may  be  assigned  to  K. 

Conies  having  their  axes  on  the  same  lines  are  said  to  be  CoaxiaL 

Hence  if  ^is  an  arbitrary  parameter,  (1)  will  represent  an  in- 
finite system  of  similar  and  coaxial  ellipses. 

For  any  particular  value  of  K  the  equations 


a 


-^=±K  (3) 


represent  a  pair  of  conjugate  hyperbolas  (§  140). 

If,  however,  iTis  arbitrary,  equations  (3)  will  give  (as  in  the 
case  of  the  ellipse)  a  system  of  similar  and  coaxial  hyperbolas, 
together  with  their  corresponding  conjugate  hyperbolas,  which 
are  also  similar.  It  follows  from  §  146  that  these  two  infinite 
systems  of  hyperbolas  all  have  the  same  asymptotes.  Moreover, 
the  asymptotes  are  the  limit  which  both  systems  approach  as  K 
becomes  zero.  Thus  two  intersecting  lines  are  not  only  one  of  a 
system  of  similar  and  coaxial  hyperbolas,  but  may  also  be  re- 
garded as  a  pair  of  self-conjugate  hyperbolas. 

It  is  also  to  be  noticed  that  although  both  axes  of  two  inter- 
secting lines  are  zero,  their  ratio  in  the  limit  is  the  tangent  of 
half  the  angle  between  the  lines. 

Cob.      The  axes  of  similar  conies  are  proportional. 


148.] 


THE   ELLIPSE   Al^D   HYPERBOLA. 


239 


148.     To  find  the  locus  of  the  middle  points  of  a  system  of  parallel 
chords  of  a  central  conic. 

Y 


I.     Let  AB  be  any  one  of  a  system  of  parallel  chords  of  the 
ellipse 

(1) 


^     I     ?/  _  IT 


Let  P(x'j  y')  be  the  middle  point  of  AB^  and  y  its  inclination 
to  the  ic-axis. 

Then  the  equation  of  AB  may  be  written  [§  46,  (4)] 

cos  Y  sin  7'  ' 

or  x=^x'  -\-  r  cos  y,     yz=y'-\-r  sin  y,  (2) 

where  r  is  the  distance  from  {x',  y')  to  any  point  (a?,  y)  on  the 
line. 

If  the  point  (ic,  ?/)  is  on  the  ellipse,  these  values  (2)  may  be 
substituted  in  equation  (1)  ;  this  gives 

{x'  +  r  cos  yY  .{y'  -{-T  sin  y)' 


K 


or 


/  cosV        sin'^  y\^,         (x^coay      y'  sin  y\ 


^==0.      (3) 


The  values  of  r  found  by  solving  this  quadratic  equation  are 
the  lengths  of  the  lines  PA  and  PB,  which  can  be  drawn  from  P 


240  THE   ELLIPSE   AND   HYPERBOLA.  [148. 

along  AB  to  the  ellipse.  Since  P  is  the  middle  point  of  the  chord, 
these  two  values  of  r  must  be  equal  in  magnitude  and  opposite  in 
sign;  i,  e.  the  sum  of  the  roots  of  (3)  must  be  zero.     Hence  (§  91) 

^^cosr   ,  ysinr_^  ... 

The  required  locus  is,  therefore,  the  straight  line 

y  =  —-^GOtr'x.  (5) 

Hence  every  diameter  (§  126)  of  an  ellipse  passes  through  the 
centre. 

Cor.  I.  All  chords  intercepted  on  the  same  line,  or  on  a  series  oj 
parallel  lines,  by  a  system  of  similar  and  coaxial  ellipses  are  bisected  by 
the  same  diameter. 

Since  equation  (5)  is  independent  of  K,  the  locus  of  P  is  the 
same  whatever  value  may  be  given  to  ^  in  (1).     (§  147.) 

Cor.  II.  If  a  straight  line  meets  each  of  two  similar  and  coaxial 
ellipses  in  two  real  points,  the  two  portions  of  the  line  intercepted  between 
them  are  equal;  i.  e.  A'A  =  BB' . 

Cor.  III.  Chords  of  an  ellipse  which  are  tangent  to  a  similar  and 
coaxial  ellipse  are  bisected  at  the  point  of  contact. 

Cor.  IV.  The  tangent  at  either  extremity  of  any  diameter  is  jmrallel 
to  the  chords  bisected  by  that  diameter. 

II.  In  like  manner,  if  y  is  the  inclination  to  the  ir-axis  of  a 
system  of  parallel  chords  of  the  hyperbolas 

g-f:  =  ±ir,  (6) 

we  find  the  locus  of  the  middle  points  of  the  chords  to  be  the 
straight  line 

y  =  -,cotr'x,  (7) 

(Jb 

for  all  values  of  K,  including  the  ease  K=0. 

Hence  all  diameters  of  a  hyperbola  pass  through  the  centre. 

The  preceding  corollaries  apply  also  to  similar  and  coaxial 
hyperbolas. 


148.] 


THE    ELLIPSE    AND    HYPEBBOLA. 


241 


Cor.  V.  Chords  intercepted  on  the  same  line,  or  on  a  system  of  'par- 
allel lines,  by  two  conjugate  hyperbolas,  and  their  asymptotes,  are  bisected 
by  the  same  diameter. 

Cor.  VI.  If  a  straight  line  meets  each  of  two  conjugate  hyperbolas 
in  real  points,  the  two  portions  of  the  line  intercepted  between  the  curves 
are  equal.  The  portions  intercepted  between  either  hyperbola  and  the 
asymptotes  are  also  equal;  i.  e.  A"A  =  BB"  and  A'A=BB'.  Hence 
the  part  of  a  tangent  to  a  hyperbola  included  between  the  two  branches  of 
its  conjugate,  and  also  the  part  included  between  its  asymptotes,  are  bi- 
sected at  the  point  of  contact. 

Ex.  1.    Find  the  locus  of  the  middle  points  of  chords  of  the  ellipse 

parallel  to  3a;  —  2y  =  l. 

Ex.  2.    Find  the  equation  of  the  chord  of  the  hyperbola 
25x2  —  163/2  =  400 
which  is  bisected  at  the  point  (5,  —  3). 

Ex.  3.    Find  the  equation  of  the  chord  of  the  ellipse  ix"^  -\-  Sy"^  =  32  which 
is  bisected  at  the  point  ( — 2,  1). 
17 


242  THE  ELLIPSE  AND   HYPERBOLA.  [149. 


Conjugate  Diameters. 

149.  We  have  seen  in  §  148  that  all  diameters  of  a  central 
conic  pass  through  the  centre.  Conversely,  every  chord  which  passes 
through  the  centre  is  a  diameter,  i.  e.  bisects  some  system  of  parallel 
chords.  For,  by  giving  y  a  suitable  value,  equations  (5)  and  (7) 
of  §  148  may  be  made  to  represent  any  chord  through  the  centre. 

If  /  is  the  inclination  to  the  a^-axis  of  the  diameter  which  bi- 
sects all  chords  whose  inclination  is  r,  we  have,  from  (5)  and  (7) 
of  §  148, 

or  tan  /  tan  /=  =h  —^  (1) 

a 

Let  y  =  mx  and  y  =  m'x  be  any  two  diameters. 
Then,  if  the  first  bisects  all  chords  parallel  to  the  second,  we 
have  from  (1) 

mm'=  -+-  -,.  (2) 

a 

Since  this  is  the  only  condition  that  must  hold  in  order  that 
the  second  may  bisect  all  chords  parallel  to  the  first,  it  follows 
that,  if  one  diameter  of  a  conic  bisects  all  chords  parallel  to  a  second,  the 
second  diameter  will  also  bisect  all  chords  parallel  to  the  first, 

Def.  Two  diameters,  so  related  that  each  bisects  every  chord 
parallel  to  the  other,  are  called  Conjugate  Diameters.* 

For  example,  the  axes  are  a  pair  of  conjugate  diameters. 

From  equation  (2)  we  see  that  the  slopes  of  two  conjugate 
diameters  of  an  ellipse  have  opposite  signs,  whereas  in  the  hyper- 
bola the  signs  are  the  same.     (See  figures  under  §  148. ) 

If  m  <  — ,  then  m'  >-,  numerically, 
a  a 

Hence  conjugate  diameters  of  an  ellipse  are  separated  by  the 

axes,  and  also  by  the  lines  ay=:±.bx]  while  conjugate  diameters 

of  a  hyperbola  are  separated  by  the  asymptotes,  but  not  by  the 

axes. 

*  It  is  evident  that  none  but  central  conies  can  have  conjugate  diameters,  since  in  the 
parabola  all  diameters  have  the  same  direction  (§  126) . 


149.]  THE  ELLIPSE  AND   HYPERBOLA.  243 

If  m  =  — ,  then  m'=: in  the  ellipse. 

a  a 

Tlie  two  diameters  are  then  equally  inclined  to  the  major  axis, 
and,  from  the  symmetry  of  the  curve,  the  two  diameters  are  equal 
in  length.  The  equations  of  the  equal  conjugate  diameters  of  an 
ellipse  are,  therefore, 

y  =  ±^-^-  (3) 

a 
If  m  =  ±  -,  then  in  the  hyperbola  m'=  ±  — ,  respectively. 

Therefore  equi-con jugate  diameters  of  a  hyperbola  coincide 
with  an  asymptote,  so  that  an  asymptote  may  be  regarded  as  a  self- 
conjugate  diameter. 

The  equi-con  jugate  diameters  of  a  conic,  therefore,  in  all  cases 
coincide  in  direction  with  the  diagonals  of  the  rectangle  formed 
by  tangents  at  the  ends  of  its  axes. 

CoR.  I.    If  two  diameters  are  conjugate  with  respect  to  one  of  two  con- 
jugate hyperbolas,  they  will  be  conjugate  with  respect  to  the  other  also. 
[(6)  and  (7),  §  148.] 

Cor.  II.  One  of  two  conjugate  diameters  q/  a  hyperbola  meets  the 
curve  in  real  points,  and  the  other  meets  the  conjugate  hyperbola  in  real 
points.     (Cor.  III. ,  §  146. ) 

For  this  reason  we  will  call  the  extremities  of  any  diameter  of 
a  hyperbola  the  points  in  which  it  cuts  either  the  primary  or  the 
conjugate  hyperbola,  as  the  case  may  be;  and  the  length  of  the 
diameter  will  be  the  distance  between  these  points. 

CoR.  III.  Tangents  at  the  ends  of  any  diameter  are  parallel  to  the 
conjugate  diameter. 

Ex.  1.    Write  down  the  equations  of  the  diameters  conjugate  to 
X  —  y  =  0f    x-\-y  =  Of    by  =  aXj    ay  =  bx. 

Ex.  2.  In  the  ellipse  2x'^  -f  %^  =  8,  find  two  conjugate  diameters,  one 
of  ^ which  bisects  the  chord  x-\-2y  =  2. 

Ex.  3.    Find  the  equation  of  the  diameter  of  the  hyperbola 

16x2  —  92/2  =  144 
conjugate  tox-\-2y  =  0. 

Ex.  4.  Find  two  conjugate  diameters  of  the  ellipse  ix^  +  25x^  =  100,  one 
of  which  passes  through  the  point  (3,  —  1). 

Ex.  5.  Find  the  equation  of  the  chord  of  the  hyperbola  x^  —  y^  =  16, 
whose  middle  point  is  ( —  2,  3). 


244 


THE   ELLIPSE    AND    HYPERBOLA. 


[150. 


1 50.     Given  the  extremity  of  any  diameter,  to  find  the  extremities  of 
the  conjugate  diameter. 


I.     Let  Pi(a?i,  2/i),  P2(^2j  2/2)  be  the  extremities  of  two  conju- 
gate diameters  of  an  ellipse. 

Then  the  equations  of  OP^  and  OP2  are 


y 


_  2/1 


-^a;     and 


^2  > 


[(1),  §149] 


«*/l«>l/o 


or 


^1^2 


0. 


(1) 


a'      '      b' 

Let  ^1,  s^2  t)e  the  eccentric  angles  of  P,,  P2,  respectively. 

Then  Xi=^a  cos  ^1,         yi  =  b  sin  ^j, 

0^2  =  a  cos  ^2j         2/2  =  ^  si^  S^2-      (§  142,  I. ) 

Substituting  these  values  in  (1),  we  have 

cos  <pi  COS  <p2  +  sin  Vi  sin  (p^  =  cos  (<pi  ^  ^2)=  0.  (2) 

.-.     s^,^^2  =  90°. 

That  is,  the  eccentric  angles  of  the  extremities  of  two  conju- 
gate diameters  of  an  ellipse  differ  by  a  right  angle.  Hence  the 
corresponding  diameters  OQi,  OQzOi  the  auxiliary  circle  are  per- 
pendicular to  one  another. 


150.] 
Since 


THE   ELLIPSE   AND   HYPERBOLA. 


245 


S^a  =  Vi  ±  90°, 

sin  ^2  =  ±  cos  <pij         cos  ^2  =  =F  sin  ^j. 
Therefore  the  extremities  of  two  conjugate  diameters  of  an 
ellipse  may  be  written 


Pj(a  cos  <pij  h  sin  ^1)   and    P2(=F  a  sin  ^j,  ±:  b  cos  ^j),  ^ 


or 


Pii^i,yi)     and    ^2(^-^?/u  ±-a^ij 


>        (3) 
I 


II.     If  Pj,  P2  are  the  extremities  of  two  conjugate  diameters 
of  a  hyperbola,  equation  (1)  becomes 


or 


0. 
a"  0' 

Then  from  §  143,  II.,  and  §  149,  Cor.  II.,  we  also  have 

Xi  =  a  sec  <pi,     yx  =  h  tan  ^1, 

0*2  ==  a  tan  ^2?     2/2  =  ^  sec  ^2* 
Substituting  these  values  in  (4)  gives 

sec  ^1  tan  ^2 —  tan  ^^  sec  ^2  =  0, 
sin  ^2  sin  <p^ 


(4) 


cos  (Ci  COS  f ,  COS  (Cj  COS  p2 

.'.     ¥'2  =  Ci,     or    JT  —  50,, 


=  0. 


(5) 
(6) 
(V) 


246  THE   ELLIPSE   AND   HYPERBOLA.  [151. 

That  is,  the  eccentric  angles  of  the  ends  of  two  conjugate 
diameters  of  a  hyperbola  are  either  equal  or  supplementary. 
Therefore  the  corresponding  diameters  0§i,  0§2  of  the  auxiliary 
circles  are  equally  inclined  to  the  transverse  axes  of  the  two  con- 
jugate hyperbolas. 

Since      tan  ^2  =  ±  tan  <p^.    and     sec  <p^-=  ±i  sec  (f^^ 

the  extremities  of  any  two  conjugate  diameters  of  a  hyperbola 
may  be  expressed  in  the  form 

Pi  (a  sec  ^1,  h  tan  ^j )    and    P<J^±.a  tan  ^1,  zb  h  sec  s^i ) ,  1 
or  Pi(a?i,  2/1)     and     F^^pj^,  ±.~x^.  \ 

151.  Tlie  sum  0/  the  squares  of  two  conjugate  semi-diameters  of  an 
ellipse  is  constant. 

Let  the  extremities  of  any  two  conjugate  diameters  be 
[§150,(3)] 

P](a  cos  ^,  b  sin  <p)     and     P2(^  ^  sin  <pj  ±b  sin  ^). 

Let  OPi  =  a',      OFi  =zb',  0  being  the  centre. 

Then  a''  =  a'  cos'  ^  +  5'  sin^  ^,  [(4) ,  §  7] 

5'^  =  a^  sin^  <P  -\-  b^  cos''  <p. 

.-.     a''-j-b''  =  a'-\-b\ 

1 52.  TA.<3  area  of  the  parallelogram  formed  by  tangents  at  the  ends 
of  conjugate  diameters  of  an  ellipse  is  constant. 

Let     Pi(a  cos  ^,  b  sin  <p)    and    PgCn^  a  sin  s^,  dz  6  cos  ^) 

be  the  extremities  of  any  two  conjugate  diameters,  and  let  ABCD 
be  the  parallelogram  formed  by  tangents  at  the  ends  of  these 
diameters. 

Draw  PjiV  perpendicular  to  OP2 ;  then 

Area  ABCD  =4:  OP^ '  P,N  =4.b''  P,N. 

Since  OP2  is  parallel  to  the  tangent  at  Pj  [§  149,  Cor.  IIL],  the 
equation  of  OP2  may  be  written  [(5),  §  143] 

3C  V 

—  cos  ^  +  r  sin  ^  =  0. 
a  0 


152.] 


THE   ELLIPSE   AND    HYPERBOLA. 

cos^  <P  +  sin^  <p  ah 


247 


ah 


I  COS"  (f       sin''  <p       Vhi^  cos^  <P  -\-  o^  sin^  ^        ^' 


Y 


Cor. 


If  angle  P1OP2  =  tOj  then 

P,N         ah 

sm  w  =  — -  =  -^ 
a'  a'h' 


EXAMPLES. 

1.  The  difference  of  the  squares  of  two  conjugate  semi-diameters  of  a 
hyperbola  is  constant. 

2.  The  area  of  the  parallelogram  formed  by  tangents  to  two  conjugate 
hyperbolas  at  the  ends  of  two  conjugate  diameters  is  equal  to  4a6. 

3.  If  w  denotes  the  angle  between  two  conjugate  diameters  of  a  hyper- 
bola, then  sin  w  =  ■    ,  , . 

4.  Show  that  the  acute  angle  between  two  conjugate  diameters  of  an 
ellipse  is  least  when  the  diameters  are  equal. 

5.  Show  that  the  eccentric  angles  of  the  extremities  of  the  equi-eon- 
jugate  diameters  of  an  ellipse  are  45°  and  135°. 

6.  Conjugate  diameters  of   a  rectangular  hyperbola  are  equal,  and 
equally  inclined  to  the  asymptotes. 

7.  Tangents  to  two  conjugate  hyperbolas  at  the  extremities  of  two  con- 
jugate diameters  meet  on  the  asymptotes. 


248  THE  ELLIPSE  AND   HYPERBOLA.  [152. 

8.  The  area  of  the  triangle  formed  by  two  semi -conjugate  diameters 
and  the  chord  joining  their  ends  is  constant. 

9.  Prove  that  for  all  values  of  m  the  line 


passes  through  the  extremities  of  two  conjugate  diameters  of  an  ellipse. 
What  is  the  corresponding  equation  for  the  hyperbola  ? 

10.  The  product  of  the  focal  radii  of  a  point  P  is  equal  to  the  square  of 
the  semi-diameter  parallel  to  the  tangent  at  P. 

11.  The  locus  of  the  poles  of  a  series  of  parallel  chords  is  the  diameter 
which  bisects  the  chords.  Hence  the  line  joining  the  intersection  of  two 
tangents  to  the  centre  bisects  the  chord  of  contact. 

12.  Find  the  equations  of  two  conjugate  diameters  of  the  hyperbola 
b^x^  —  0^2/2  =  aW,  one  of  which  bisects  the  chord  through  (0,  b)  and  (ae,  0). 

13.  In  the  hyperbola  4x-  —  by"^  =  20  find  the  equations  of  two  conjugate 
diameters,  one  of  which  bisects  the  chord  2x-}-Sy  =  6. 

14.  If  straight  lines  dra-wn  through  any  point  of  an  ellipse  to  the  ends 
of  any  diameter  POP^  meet  the  conjugate  diameter  PiOP/  in  Q  and  R,  show 
that  OQ' OR  =  OPi\ 

15.  Show  that  the  locus  of  the  intersection  of  the  perpendiculars  from 
the  foci  upon  a  pair  of  conjugate  diameters  of  an  ellipse  is  a  similar  con- 
centric ellipse. 

16.  Two  conjugate  diameters  of  an  ellipse  are  drawn,  and  their  four  ex- 
tremities are  joined  to  any  point  on  a  given  circle  whose  centre  is  at  the 
centre  of  the  ellipse.  Show  that  the  sum  of  the  squares  of  these  four  lines 
is  constant. 

17.  Any  tangent  to  an  ellipse  meets  the  director  circle  in  P  and  Q, 
Prove  that  OP  and  OQ  are  semi -conjugate  diameters  of  the  ellipse. 

18.  Pi  is  a  point  on  a  branch  of  a  hyperbola,  P2  is  a  point  on  a  branch 
of  its  conjugate,  OPi  and  OPy  being  semi-conjugate  diameters.  If  Pi  and 
P2  are  the  interior  foci  of  these  two  branches,  respectively,  show  that 

F,P,  ~  F,P,  =  a^b. 

19.  Find  the  equation  of  the  chord  passing  through  the  extremities  of 
two  conjugate  diameters. 

20.  The  lengths  of  the  chords  joining  the  extremities  of  two  conjugate 
diameters  of  an  ellipse  are 

l/a^+b^zbaV  sin29). 

For  what  value  of  (p  are  these  chords,  one  a  maximum  and  the  other  a 
minimum  ? 
Show  that  the  corresponding  result  for  the  hyperbola  is 

ae(sec  (j>  zt  tan  ^). 


153.]  THE   ELLIPSE  AND   HYPERBOLA.  249 

21.  If  the  eccentric  angles  of  two  points  P,  Q  on  an  ellipse  are  ^i,  02, 
prove  that  the  area  of  the  parallelogram  formed  by  tangents  at  the  ends  of 
diameters  through  P  and  Q  is 

idb  csc(«;&i  —  02) ; 

and  hence  show  that  this  area  is  least  when  Pand  Q  are  the  ends  of  con- 
jugate diameters. 

22.  The  sides  of  a  parallelogram  circumscribing  an  ellipse  are  parallel 
to  conjugate  diameters ;  prove  that  the  product  of  the  perpendiculars  from 
two  opposite  vertices  on  any  tangent  is  equal  to  the  product  of  those  from 
the  other  two  vertices. 

23.  The  radius  of  a  circle  which  touches  a  hyperbola  and  its  asymptotes 
is  equal  to  that  part  of  the  latus  rectum  intercepted  between  the  curve  and 
the  asymptotes. 

24.  A  line  parallel  to  the  y-axis  meets  two  conjugate  hyperbolas  and  one 
of  their  asymptotes  in  P,  Q^R,  Show  that  the  normals  at  P,  Q^  R  meet  on 
the  X-axis. 

25.  If  a  tangent  drawn  at  any  point  Pof  the  inner  of  two  similar  coaxial 
conies  meets  the  outer  in  the  points  T  aud  T^,  then  any  chord  of  the  inner 
through  P  is  half  the  algebraic  sum  of  the  parallel  chords  of  the  outer 
through  rand  T\ 

26.  Def.  The  two  chords  of  a  central  conic  which  join  any  point  on  the 
curve  to  the  extremities  of  any  diameter  are  called  Supplemental  Chords. 

Show  that  two  supplemental  chords  are  parallel  to  a  pair  of  conjugate 
diameters. 

153.  To  find  the  equation  of  a  conic  referred  to  a  pair  of  conjugate 
diameters  as  axes. 

We  may  assume  the  required  equation  to  be  (§67) 

ax'.+  2hxy  +  bi/-{'2gx  +  2fy-^c  =  0.  (1) 

By  supposition  each  axis  bisects  all  chords  parallel  to  the  other 
(§  149).  Hence  for  every  value  of  either  x  or  y  the  two  values 
of  the  other  must  be  equal  in  magnitude  and  opposite  in  sign. 

Solving  (1)  for  £c  and  y  respectively  gives 

ax  =  -(hy  +  g)±^.  (2) 

and  by  =  -  Qix  +/)  ±  ^';  (3) 

so  that  for  the  values  of  ic  to  be  opposite  hy-^-g  must  vanish  for 
all  values  of  y,  while  for  the  values  of  y  to  be  opposite  hx  +/  must 


250  THE   ELLIPSE   AND   HYPERBOLA.  [153. 

vanish  for  all  values  of  x.     That  is,  /^  ^  =  /i  =  0,  and  we  have 

aicH%'  +  c-0,  '  (4) 

4+4=1.  C5) 

a  b 

Since  (a',0)  and  (0,6')  are  points  on  the  curve 

a''  =  --     and     b''  =  -~ 
a  0 

Therefore  the  equation  of  the  ellipse  referred  to  conjugate 
diaijaeters  is 

5  +  6-  =  !'  w 

where  a',  b'  are  the  lengths  of  the  semi-diameters. 

In  the  case  of  the  hyperbola,  the  conic  meets  one  of  the  axes  in 
imaginary  points ;  let  this  be  the  ?/-axis.     Then  the  equation  is 

a"      b"  ^'^ 

The  conjugate  hyperbola  will  therefore  meet  the  a^-axis  in 
imaginary  points  (§  149,  Cor.  III.))  so  that  its  equation  is 

|.-5  =  1.  (8) 

where  a',  b'  are  the  real  intercepts  of  the  two  curves  (7)  and  (8) 
on  the  axes. 

If  the  ellipse  is  referred  to  its  equi -con jugate  diameters,  so  that 
a'=b',  its  equation  will  be 

x'+f  =  a'\  (9) 

We  thus  see  that  the  equation  of  a  conic  referred  to  a  pair  of 
conjugate  diameters  is  of  the  same  form  as  when  referred  to  its 
own  axes.  Moreover,  the  proofs  in  §  HI,  §  113,  and  §  139  hold 
good  whether  the  axes  are  rectangular  or  oblique.  Therefore, 
when  the  equation  of  the  conic  is  referred  to  a  pair  of  conjugate 
diameters,  the  equations  of  the  polar  and  tangent  will  be  of  the 
same /orm  as  those  given  in  §§  138,  139. 


154.]  THE  ELLIPSE  AND   HYPERBOLA.  251 

1 54.     To  find  the  equation  of  a  hyperbola  when  referred  to  its  asymp- 
totes as  axes  of  coordinates. 

The  equations  of  two  conjugate  hyperbolas  referred  to  their 
own  axes  aie 

^+|-=±i.  (1) 

Let  io  be  the  angle  between  the  asymptotes  such  that 

tan|=^.  (§U6.) 


Then         sin-=       '  ,        cos -  =  --==-.  (2) 


a 

Vo, 

6 

l/o 

'+6^    J 

Since  the  axes  bisect  the  angles  between  the  asymptotes,  the 
formulae  for  effecting  the  required  transformation  are,  from  (2 ) 
and  Ex.  20,  p.  98, 

V  a"  -(-  ft' 

Substituting  these  values  in  (1),  reducing  and  dropping  the 
accents,  we  obtain 

40^2/ =±  (a' +  60,  (4) 

which  is  the  required  equation. 

If  the  hyperbola  is  rectangular,  so  that  a  =  b,  its  equation  re- 
ferred to  its  asymptotes  will  be 

2xy=±a\  (5) 

Otherwise.  The  equation  of  the  asymptotes,  referred  to  them- 
selves as  axes  of  coordinates^  is  xy  ==  0. 

Therefore  the  equations  of  any  two  conjugate  hyperbolas  re- 
ferred to  them  is  of  the  form  (§  146) 

xy=±K.  (6) 

Hence  the  equation  xy  =  K,  where  K  is  any  constant,  always 
represents  a  hyperbola  referred  to  its  asymptotes  as  axes  of  co- 
ordinates ;  so  that,  if  the  axes  of  coordinates  are  at  right  angles, 
the  hyperbola  xy  =  K  is  rectangular. 


262  THE   ELLIPSE   AND   HYPERBOLA.  [155. 

165.  To  find  the  equation  of  the  tangent  at  any  point  (x',  y')  of 
the  hyperbola  xy  =^  K. 

Since  equation  (6),  §  111,  holds  good  for  oblique  axes,  the  re- 
quired equation  may  be  written 

xy'  -\-  x'y  —  2jfir, 

X    .y        2K 

or  — r+    /=— 7-7-  (1) 

x'        y'        x'y'  ^ 

Since  (a?',  i/')  is  on  the  curve,  x*y'=^K^  and  therefore  the 
equation  of  the  tangent  at  (a?',  ?/')  is 

-^  +  4  =  2.  (2) 

x'        y'  ^   ^ 

Cor.  The  area  of  the  triangle  formed  hy  the  asymptotes  and  any 
tangent  to  a  hyperbola  is  constant. 

From  equation  (2)  we  see  that  the  intercepts  of  any  tangent 
on  the  asymptotes  are  2x'  and  2y'. 

Hence  if  A  denotes  the  ai-ea,  we  have 

A  =  2x'y'  sin  (o. 

10        to  2ab 

But  sin«>  =  2sin-cos-  =  -^^-pp,    [§154,(2)] 

and  2x'y'  =  i(a^  +  b').  [§  154,  (4)] 

,*.     A  =  ab. 

Ex.  1.  Show  that  the  tangent  to  the  interior  of  any  two  similar  and  co- 
axial conies  cuts  off  a  constant  area  from  the  exterior  conic. 

Ex.  2.  Show  that  the  equation  of  the  normal  to  the  rectangular  hyper- 
bola x^  —  y"^  =  a^  at  the  point  (x^,  y^)  may  be  written 

|/+|>-2.  (C/.  Ex.  22,  p.  122.) 

156.     The  parabola  is  a  limiting  form  of  each  of  the  central  conies. 

The  equation  of  the  central  conic  referred  to  the  left  vertex  as 
origin  is  found  by  writing  (^x  —  a)  for  x  in  the  standard  equation. 
The  equation  thus  obtained  w411  be 

£;±i;_^=o,  (1) 

f'±^-2x  =  0.  (2) 


157.]  THE   ELLIPSE   AND   HYPERBOLA.  253 

Let  —  =1  =  half  the  latus  rectum  (§  138),  then 
a 

f±^-2^  =  0.  (3) 

Now  if  a  becomes  infinite,  while  I  remains  finite,  the  equation 
(3)  becomes  in  the  limit 

y'=±2lx,  (4) 

which  is  the  equation  of  a  parabola,  whether  we  use  the  plus  or 

minus  sign  in  the  second  member. 

b^ 
Since  Z  =  — ,  it  follows  that  if  I  is  not  zero  when  a  is  infinite,  b 
a 

must  also  be  infinite.     Therefore  the  parabola  is  the  limiting 

form  of  the  central  conic  whose  latus  rectum  is  finite,  but  whose 

axes  are  both  infinite,  the  centre  and  one  focus  being  at  infinity. 

Notice  that,  although  both  a  and  b  are  infinite,  a  is  infinite 

compared  to  6,  in  fact  t  =  -j'     Thus  the  vddth  of  a  parabola  is 

nothing  compared  to  its  length,  and  we  see  how  coincident  or 
parallel  lines  are  limiting  cases  of  parabolas.  The  shape  is  the 
same,  only  the  scale  has  been  changed. 

1 57.     To  find  the  polar  equation  of  a  central  conic,  the  pole  being  at 
the  centre. 

The  formulae  for  changing  from  rectangular  to  polar  coordinates 

are  (§6) 

x==  p  cos  Oj     y  =  P  sin  0. 

These  values  substituted  in 
give  p=^^_^_+_^j=l, 


or  p^ 


±a'b'  __  ±a'b^ 

a^  sin'  O±zb^coa  e^a'—  (a'  =F  b') cos^  $ 

•  •     '^~~l-e=^co8»^' 


which  is  the  require^  equation. 


254  THE   ELLIPSE   AJS^D    HYPERBOLA.  [157. 


Examples  on  Chapter  X. 

1.  Qis  the  point  on  the  auxiliary  circle  corresponding  to  P  on  the 
ellipse ;  PLM  is  drawn  parallel  to  0  ^  *  to  meet  the  axes  in  L  and  M.  Prove 
that  PL  =  b  and  PM=a. 

2.  Show  that  the  sum  of  the  squares  of  the  reciprocals  of  two  perpen- 
dicular diameters  of  an  ellipse  is  constant.    (See  §  156.) 

3.  If  an  equilateral  triangle  is  inscribed  in  an  ellipse,  the  sum  of  the 
squares  of  the  reciprocals  of  the  diameters  parallel  to  the  sides  is  constant. 

4.  Find  the  inclination  to  the  major  axis  of  the  diameter  of  an  ellipse 
the  square  of  whose  length  is  (1)  the  arithmetical  mean,  (2)  the  geometri- 
cal mean,  and  ^3)  the  harmonical  mean  between  the  squares  on  the  major 
and  minor  axes.    (§  156.)  Ans.  to  (3).    45°. 

5.  In  a  rectangular  hyperbola  the  aiigles  subtended  at  its  vertices  by 
any  chord  parallel  to  its  conjugate  axis  are  supplementary. 

6.  In  a  rectangular  hyperbola  the  angle  subtended  by  any  chord  at  the 
centre  is  the  supplement  of  the  angle  between  the  tangents  at  the  ends  of 
the  chord. 

7.  Any  point  P  of  an  ellipse  is  joined  to  the  extremities  of  the  major 
axis;  prove  that  the  portion  of  a  directrix  intercepted  between  the  joining 
lines  subtends  a  right  angle  at  the  corresponding  focus. 

8.  The  normal  to  the  hyperbola  b'^x^  —  a'^y^  =  a^b^  meets  the  axes  in 
M,  N,  and  MP  and  iVP  are  drawn  perpendicular  to  the  axes;  prove  that  the 
locus  of  P  is  the  similar  hyperbola 

a^x'^  —  bV  =  («'  +  &')'. 

9.  The  tangent  at  any  point  P  of  an  ellipse  meets  the  tangent  at  one 
vertex  in  Q;  show  that  OQ  is  parallel  to  the  line  joining  Pto  the  other 
vertex. 

10.  Prove  that  the  foci  of  the  hyperbola  ^xy  =  a^  -|-  6^  are  given  by 

11.  Show  that  two  concentric  rectangular  hyperbolas  whose  axes  meet 
at  an  angle  of  45°  cut  orthogonally. 

12.  A  point  moves  so  that  the  sum  of  the  squares  of  its  distances  from 
two  intersecting  straight  lines  is  constant.  Prove  that  its  locus  is  an 
ellipse,  and  find  the  eccentricity  in  terms  of  the  angle  between  the  lines. 

13.  PNP^  is  a  double  ordinate  of  an  ellipse,  and  Q  is  any  point  on  the 
curve ;  show  that  if  QP  and  QP^  meet  the  major  axis  in  M  and  M^,  re- 
spectively, 

OM'OM'  =  a\ 

*  In  this  set  of  examples  O  is  always  at  the  centre  of  the  conic. 


.157.]  THE  ELLIPSE   AND   HYPERBOLA.  255 

14.  A  straight  line  always  passes  through  a  fixed  point;  prove  that  the 
locus  of  the  middle  point  of  the  portion  of  it,  which  is  intercepted  between 
two  fixed  straight  lines,  is  a  hyperbola  whose  asymptotes  are  parallel  to 
the  fixed  lines. 

15.  A  straight  line  has  its  extremities  on  two  fixed  straight  lines  and 
cuts  off  from  them  a  triangle  of  constant  area.  Find  the  locus  of  the 
middle  point  of  the  line. 

16.  The  tangent  to  an  ellipse  at  P  meets  the  axes  in  T,  T\  and  OQ  is  the 
perpendicular  on  it  from  the  centre.  Prove  (1)  that  TT^ 'PQ  =  a^  —  b^, 
and  (2)  that  the  least  value  of  TT^  is  a  +  6. 

17.  Show  that  the  four  lines  which  join  the  foci  to  two  points  Pand  Q 
on  an  ellipse  all  touch  a  circle  whose  centre  is  the  pole  of  PQ. 

18.  If  the  ordinate  NP  of  any  point  P  of  any  ellipse  is  produced  to  Q,  so 
that  NQ  is  equal  to  the  subtangent  at  P,  prove  that  the  locus  of  ^  is  a 
hyperbola. 

19.  The  rectangular  coordinates  of  a  point  are  a  tan  (6  -\-  a)  and 
b  tan  (^  +  /3)>  where  ^  is  variable;  prove  that  the  locus  of  the  point  is  p. 
hyperbola. 

20.  The  tangent  at  any  point  P  of  an  eHipse  meets  the  equi-conjugate 
diameters  in  Q,  Q^;  show  that  the  triangles  POQ  and  POQ^  are  in  the  ratio 
of  OQ"  :  OQ'K 

21.  If  in  an  ellipse  0$  is  conjugate  to  the  normal  at  P,  then  will  OP  be 
conjugate  to  the  normal  at  Q. 

22.  OA  and  OB  are  fixed  straight  lines,  Pany  point,  PM  and  PN  the 
perpendiculars  from  P  on  OA  and  OB.  If  the  area  of  the  quadrilateral 
OMPN  is  constant,  show  that  by  a  proper  choice  of  axes  the  locus  of  P  is 

x^  —  y^  =  KcsoAOB. 

23.  A  series  of  circles  touches  a  given  straight  line  at  a  given  point. 
Prove  that  the  locus  of  the  poles  of  another  fixed  straight  line  with  regard 
to  these  circles  is  a  hyperbola  whose  asymptotes  are,  respectively,  a  parallel 
to  the  first  given  line  and  a  perpendicular  to  the  second. 

24.  A  point  is  such  that  the  perpendicular  from  the  centre  on  its  polar 
with  respect  to  the  ellipse  is  constant  and  equal  to  c.  Show  that  its  locus 
is  the  ellipse 

25.  If  any  pair  of  conjugate  diameters  of  an  ellipse  cut  the  tangent  at  a 
point  Pin  Tand  T',  show  that  TP-PT=  0P'\  where  OP'  is  the  diameter 
conjugate  to  OP.    (§  153.) 

26.  If  a  number  of  hyperbolas  have  the  same  transverse  axis,  show  that 
tangents  at  points  having  a  common  abscissa  meet  the  aj-axis  in  the  same 


256  THE    ELLIPSE    AND    HYPERBOLA.  [157. 

point ;  and  that  tangents  to  the  conjugate  hyperbolas  at  points  having  the 
same  abscissa  meet  the  same  axis  in  a  point  equally  distant  from  the 
centre. 

27.  If  the  focus  of  an  ellipse  is  the  common  focus  of  two  parabolas, 
whose  vertices  are  at  the  ends  of  the  major  axis,  these  parabolas  will  inter- 
sect at  right  angles  in  points  P  and  Q  such  that 

28.  If  P,  P'  are  the  extremities  of  conjugate  diameters  of  an  ellipse,  and 
FQ,  PQ^  are  chords  parallel  to  an  axis  of  the  ellipse,  show  that  PQ^  and 
P^Q  are  parallel  to  the  equi -conjugate  diameters. 

29.  The  two  straight  lines  joining  the  points  in  which  two  tangents  to 
a  hyperbola  meet  the  asymptotes  are  parallel  to  the  chord  of  contact  of 
the  tangents  and  equidistant  from  it.     (§  154.) 

30.  If  through  any  point  P  a  line  PQR  is  drawn  parallel  to  an  asymp- 
tote of  a  hyperbola,  cutting  the  curve  in  Q  and  the,  polar  of  P  in  P,  show 
that 

PQ=QR.  [Use  §46,  (4).] 

31.  Prove  that  the  ends  of  the  latera  recta  of  all  ellipses,  having  a  given 
major  axis  2a,  lie  on  the  parabolas  x^  ±ay  =  0^. 

32.  The  tangent  at  any  point  Pof  a  circle  meets  the  tangent  at  a  fixed 
point  A  in  T,  and  T  is  joined  to  P,  the  other  end  of  the  diameter  through  A. 
Prove  that  the  locus  of  the  intersection  of  -4Pand  PTis  an  ellipse  whose 
eccentricity  is  ^v/2. 

33.  Prove  that  the  part  of  the  tangent  at  any  point  of  a  hyperbola  in- 
tercepted between  the  point  of  contact  and  the  transverse  axis  is  a  har- 
monic mean  between  the  lengths  of  the  perpendiculars  drawn  from  the  foci 
on  the  normal  at  the  same  point.    (See  footnote  to  §  115.) 

34.  If  a  right  triangle  is  inscribed  in  a  rectangular  hyperbola,  the  tan- 
gent at  the  vertex  of  the  right  angle  is  perpendicular  to  the  hypotenuse. 

35.  If  P,  P^  are  the  extremities  of  conjugate  diameters,  and  the  tangent 
at  P  cuts  the  major  axis  in  T,  and  the  tangent  at  P^  cuts  the  minor  axis  in 
T\  show  that  TT^  will  be  parallel  to  one  of  the  equi -conjugates. 

36.  QQ^  is  any  chord  of  an  ellipse  parallel  to  one  of  the  equi-conjugates, 
and  the  tangents  at  Q,  Q^meet  in  T.  Show  that  the  circle  QT^^  passes 
through  the  centre.    (§153.) 

37.  If  one  axis  of  a  variable  central  conic  is  fixed  in  magnitude  and 
position,  prove  that  the  locus  of  the  point  of  contact  of  a  tangent  drawn 
from  a  fixed  point  on  the  other  axis  is  a  parabola. 

38.  Find  the  coordinates  of  the  points  of  contact  of  the  common  tan- 
gents to  the  two  hyperbolas 

and    xy  =  2a2. 


157.]  THE   ELLIPSE   AND    HYPERBOLA.  257 

39.  PNP^is  a  double  ordinate  of  an  ellipse,  and  the  normal  at  P  meets 
OP"  in  Q,    Show  that  the  locus  of  $  is  a  similar  ellipse. 

40.  In  an  ellipse  the  normal  at  P  meets  the  major  axis  in  N  and  the 
minor  axis  in  N\  Show  that  the  loci  of  the  middle  points  of  PN  and  PN^ 
are,  respectively,  the  ellipses 

41.  Show  that  the  line  y  =  mx-\-2cV  —  m  always  touches  the  hyper- 
bola xy  =  c^f  and  that  its  point  of  contact  is 


(7=S'  "i^-™)- 


42.  A  point  P  moves  along  the  fixed  line  y  =  mx;  prove  that  the  locus 
of  the  foot  of  the  perpendicular  from  P  on  its  polar  with  respect  to  the 
ellipse  b^x^  -}-  <^^y^  =  ^^^^  is  a  rectangular  hyperbola,  one  of  whose  asymp- 
totes is  the  diameter  of  the  ellipse  conjugate  to  the  given  fixed  line. 

43.  The  tangents  drawn  from  a  point  P  to  an  ellipse  make  angles  ji  and 
72  with  the  major  axis.  Find  the  locus  of  Pwhen  (1)  tan /i-f-tan  72is 
constant,  (2)  cot  yi  +  cot  y^  is  constant,  and  (3)  tan  71  tan  72  is  constant. 

44.  The  line  joining  two  extremities  of  any  two  diameters  of  an  ellipse 
is  either  parallel  or  conjugate  to  the  line  joining  two  extremities  of  their 
conjugate  diameters. 

45.  Af  A'  are  the  vertices  of  a  rectangular  hyperbola,  and  P  is  any  point 
on  the  curve ;  show  that  the  internal  and  external  bisectors  of  the  angle 
APA'  are  parallel  to  the  asymptotes. 

46.  Ay  A'  are  the  ends  of  a  fixed  diameter  of  a  circle,  and  PP'  is  any 
chord  perpendicular  to  AA' .  Show  that  the  locus  of  the  intersection  of  ^P 
and  A'P'  is  a  rectangular  hyperbola.  Show  also  that  the  words  drcXe  and 
rectangular  hyperbola  can  be  interchanged. 

47.  If  P  and  P^  are  the  ends  of  conjugate  diameters  of  an  ellipse,  show 
that  (1)  the  tangents  at  Pand  P'  meet  on  the  ellipse  ^  +  r-j  =  2,  {2)  the 

x^      1/"      1 
locus  of  the  middle  point  of  PP^  is  "2  + 12  =  o"»  ^^^  ^^^  *^®  locus  of  the 

foot  of  the  perpendicular  from  the  centre  upon  PP^  is 

48.  A  line  is  drawn  parallel  to  the  minor  axis  of  an  ellipse  midway  be- 
tween a  focus  and  the  corresponding  directrix ;  prove  that  the  product  of 
the  perpendiculars  upon  it  from  the  ends  of  any  chord  passing  through 
that  focus  is  constant. 

49.  Show  that  the  coordinates  of  the  point  of  intersection  of  two  tan- 
gents to  the  hyperbola  xy  =  Ksltb  harmonic  means  between  the  coordinates 
of  the  points  of  contact.    (See  footnote  to  §  115.) 

18 


258  THE   ELLIPSE   AND    HYPERBOLA.  [158. 

50.  From  any  point  on  one  hyperbola  tangents  are  drawn  to  another 
hyperbola  which  has  the  same  asymptotes.  Show  that  the  chord  of  con- 
tact cuts  off  a  constant  area  from  the  asymptotes.    (See  Ex.  22,  p.  233.) 

51.  If  the  chord  joining  two  points  whose  eccentric  angles  are  a  and  ^ 
cuts  the  major  axis  of  an  ellipse  at  a  distance  d  from  the  centre,  show  that 

tan  s-  tan  ^  =  -^—. — . 
2         2        d-\-a 

52.  If  any  two  chords  are  drawn  through  two  points  on  the  major  axis  of 
an  ellipse  equidistant  from  the  centre,  show  that  tan  q-  tan  S- tan  ^  tan^-  =  1, 

u  Z  ^  C 

where  a,  /?,  7,  6  are  the  eccentric  angles  of  the  ends  of  the  chords.    (Use 
Ex.  51.) 

53.  If  jP,  F^  are  the  foci  of  an  ellipse  and  any  point  A  is  taken  on  the 
curve  and  chords  AFB,  BF^Cy  CFDy  DF^E  ...  are  drawn,  and  the  eccen- 
tric angles  of  A,  B,  C,  D  »  ,  ,  are  ^1,  O2, 6*3,  ^4  ...  ,  prove  that 

tan|tan^  =  cot^cot^  =  tan^tan^=  .  .  .  (Use  Ex.  52.) 

54.  Show  that  the  locus  of  the  poles  with  respect  to  the  parabola  y"^  =  iax 
of  tangents  to  the  hyperbola  x*  —  y^  =  a^  is  the  ellipse  ^x^-\-y'^  =  ia^. 

55.  Show  that  the  locus  of  the  pole,  with  respect  to  the  auxiliary  circle, 
of  a  tangent  to  the  ellipse  is  a  similar  concentric  ellipse,  whose  major  axis  is 
perpendicular  to  that  of  the  original  ellipse. 

56.  Prove  that  the  locus  of  the  pole,  with  respect  to  the  ellipse,  of  any 
tangent  to  the  auxiliary  circle  is  the  curve 

57.  Chords  of  the  ellipse  touch  the  parabola  ay"^  =  —  2b^x.  Prove  that 
the  locus  of  their  poles  is  the  parabola  ay^  =  2b^x, 

58.  Show  that  the  pole  of  any  tangent  to  the  rectangular  hyperbola 
xy  =  c^f  with  respect  to  the  circle  x^-^y^  =  a\  lies  on  a  concentric  and 
similarly  placed  rectangular  hyperbola. 

59.  Tangents  are  drawn  from  any  point  on  the  ellipse  6 V  -|-  a^y"^  =  a^h^ 

to  the  circle  x"^ -\- y"^  —  r"^ ;  prove  that  the  chords  of  contact  are  tangents  to 
the  ellipse  d^x^  -\-  b'^y^  =  r^. 

If  ^  =  — 2  +  o>  prove  that  the  lines  joining  the  centre  to  the  points  of 
contact  with  the  circle  are  conjugate  diameters  of  the  second  ellipse. 

60.  Prove  that  the  locus  of  the  pole  with  respect  to  the  hyperbola 
6V  —  dy"^  =  aW  of  any  tangent  to  the  circle,  whose  diameter  is  the  line 
joining  the  foci,  is  the  ellipse 

x\y'  _      1 


157.]  THE   ELLIPSE   AND   HYPEBBOLA.  259 

61.  In  the  two  ellipses 

%+t  =  l    and    |'  +  |'  =  a+5 

tangents  to  the  former  meet  the  latter  in  Pand  Q.    Prove  that  the  tan- 
gents at  P  and  Q  are  at  right  angles  to  each  other. 

62.  A  tangent  to  the  parabola  x^  =  iay  meets  the  hyperbola  xy  =  c'^m 
two  points  P  and  Q.    Prove  that  the  middle  point  of  PQ  lies  on  a  parabola. 

63.  From  points  on  the  circle  x^-\-y''  =  a^  tangents  are  drawn  to  the 
hyperbola  x^  —  y^  =  a^;  prove  that  the  locus  of  the  ruiddle  points  of  the 
chords  of  contact  is  the  curve 

(ix'-y'y  =  a\x^+y'). 

64.  Show  that  the  locus  of  the  poles  of  normal  chords  of  an  ellipse  is 
the  curve 

(a2  _  b^yxY  =  aY  +  b'a;2. 

65.  Prove  that  the  locus  of  the  poles  of  all  normal  chords  of  the  rect- 
angular hyperbola  xy  =  &  is  the  curve 

(»'  — 2^')'  +  4c2a;y  =  0. 

66.  P,  Q  are  fixed  points  on  an  ellipse  and  B,  any  other  point  on  the 
curve;  F,  F'are  the  middle  points  of  PB,  Q7i\  and  FG,  V^G^  Are  perpen- 
dicular to  PR  and  QR,  respectively,  and  meet  the  axis  in  G,  O^,  Show  that 
GGMs  constant. 

67.  On  the  focal  radii  of  any  point  of  an  ellipse  as  diameters  two  cir- 
cles are  described;  prove  that  the  eccentric  angle  of  the  point  is  equal  to 
the  angle  which  a  common  tangent  to  the  circles  makes  with  the  minor 
axis.    (See  Ex.  43,  p.  206). 

68.  Af  A'  are  the  vertices  of  an  ellipse,  and  P any  point  on  the  curve; 
show  that,  if  PiV  is  perpendicular  to  ^Pand  PM  perpendicular  to  A'P^  M 
and  N  being  on  the  axis  AA\  then  will  MN  be  equal  to  the  latus  rectum. 

69.  The  equation  of  the  line  joining  the  two  points  (xi,  2^1)  and  (o^  2/2) 
on  the  hyperbola  xy  =  c^  is 

»l+iC2         2/1  +  2/2 

70.  Show  that  the  area  of  the  triangle  formed  by  the  tangents  to  an 
ellipse  at  the  points  whose  eccentric  angles  are  a,  /3,  y,  respectively,  is 

ab  tan  ^(^a  —  /?)  tan  ^(/3  —  y)tanj(y  — a). 

71.  If  P,  P^  are  the  points  of  contact  of  perpendicular  tangents  to  an 
ellipse,  and  §,  Q^are  the  corresponding  points  on  the  auxiliary  circle,  show 
that  OQ  and  OQ^  are  conjugate  diameters  of  the  ellipse. 

72.  The  locus  of  the  point  from  which  can  be  drawn  two  straight  lines 
at  right  angles  to  each  other,  each  of  which  touches  one  of  the  rectangular 


260  THE  ELLIPSE  AND  HYPERBOLA.  [157. 

hyperbolas  xy  =  ±:  c^,  is  also  the  locus  of  the  feet  of  the  perpendiculars 
from  the  origin  on  the  tangents  to  the  rectangular  hyperbolas 


73.  Show  that  the  locus  of  the  intersection  of  two  tangents  to  an  ellipse, 
if  the  sum  of  the  eccentric  angles  of  the  points  of  contact  is  equal  to  the 
constant  2a,  is  the  line  ay  =  6x  tan  a. 

74.  If  the  difference  of  the  eccentric  angles  of  the  points  of  contact  of 
two  tangents  to  an  ellipse  is  120°,  show  that  the  tangents  will  intersect  on 
the  ellipse 

„2  ^  52  -  *• 

75.  Show  that  the  locus  of  the  middle  points  of  chords  of  an  ellipse 
which  pass  through  the  point  (^,  k)  is 

h'xix  —  h)-\-  a^y(y  —  fc)  =  0. 

76.  A  parallelogram  is  constructed  with  its  sides  parallel  to  the  asymp- 
totes of  a  hyperbola,  and  one  of  its  diagonals  is  a  chord  of  the  hyperbola ; 
show  that  the  other  diagonal  will  pass  through  the  centre. 

77.  Two  equal  circles  touch  one  onother ;  find  the  locus  of  a  point  which 
moves  so  that  the  sum  of  the  tangents  from  it  to  the  two  circles  is  constant. 

78.  Prove  that  the  sum  of  the  products  of  the  perpendiculars  from  the 
two  extremities  of  each  of  two  conjugate  diameters  on  any  tangent  to  an 
ellipse  is  equal  to  the  square  of  the  perpendicular  from  the  centre  on  that 
tangent. 

79.  The  straight  lines  drawn  from  any  point  on  an  equilateral  hyperbola 
to  the  extremities  of  any  diameter  are  equally  inclined  to  the  asymptotes. 

80.  -  If  the  product  of  the  perpendiculars  from  the  foci  upon  the  polar  of 
■  P,  with  respect  to  the  ellipse,  is  constant  and  equal  to  c^,  prove  that  the 

locus  of  P  is  the  ellipse 

b^x\c^  +  a^e^)  +  a^cY  =  a^bK 

81.  Prove  that  the  locus  of  the  middle  points  of  the  portions  of  tangents 
to  an  ellipse  included  between  the  axes  is  the  curve 

x'^y' 

82.  The  locus  of  the  middle  points  of  normal  chords  of  the  rectangular 
Tiyperbola  x"^  —  y^  =  a^  is  the  curve 

(^2/2  — x2)3  =  4a2xV. 

83.  From  the  centre  O  of  two  concentric  circles  two  radii  OQ,  OR  are 
drawn  equally  inclined  to  a  fixed  straight  line,  the  first  to  the  outer  circle, 
the  second  to  the  inner.  Prove  that  the  locus  of  the  middle  point  P  of  QR 
is  an  ellipse,  that  PQ,  is  the, normal  at  P  to  this  ellipse,  and  that  QR  is 
equal  to  the  diameter  conjugate  to  OP. 


157.]  THE  ELLIPSE  AND   HYPERBOLA.  261 

84.  The  locus  of  the  intersection  of  nonnals  to  an  ellipse  at  the  ends  of 
conjugate  diameters  is  the  curve 

85.  If  two  tangents  to  a  conic  are  at  right  angles  to  one  another,  the 
product  of  the  perpendiculars  from  the  centre  and  the  intersection  of  the 
tangents  on  the  chord  of  contact  is  constant. 

86.  A  circle  intersects  a  hyperbola  in  four  points.  Prove  that  the  prod- 
uct of  the  distances  of  the  four  points  of  intersection  from  one  asymptote 
is  equal  to  the  product  of  their  distances  from  the  other. 

87.  Show  that  if  a  rectangular  hyperbola  cuts  a  circle  in  four  points 
the  centre  of  mean  position  of  the  four  points  is  midway  between  the 
centres  of  the  two  curves.    (See  Ex.  33,  p.  17.) 

88.  Qish  point  on  the  normal  at  any  point  P  of  an  ellipse  such  that  the 
lines  OF  and  OQ  make  equal  angles  v^ith  the  axis  of  the  ellipse.  Show 
that  FQ  varies  as  the  diameter  conjugate  to  OP, 

89.  If  P  is  any  point  on  an  ellipse  and  any  chord  PQ  cuts  the  diameter 
conjugate  to  OP  in  i2,  then  will  PQ '  PR  be  equal  to  half  the  square  on  the 
diameter  parallel  to  PQ, 

90.  Show  that  the  locus  of  the  middle  points  of  all  chords  of  an  ellipse 
which  are  of  constant  length,  2c,  is 

(i:+i-:-oa:+i:)=w(^:+i:>("-(4).5*6.) 

91.  Tangents  at  right  angles  are  drawn  to  an  ellipse.  Show  that  the 
locus  of  the  middle  point  of  the  chord  of  contact  is  the  curve 


92.  The  area  of  the  parallelogram  formed  by  the  tangents  at  the  ends 
of  any  two  diameters  of  an  ellipse  varies  inversely  as  the  area  of  the  par- 
allelogram formed  by  joining  the  points  of  contact. 

93.  POP^  is  a  diameter  of  an  ellipse,  QOQ^  the  corresponding  diameter 
of  the  auxiliary  circle,  and  (j>  the  eccentric  angle  of  P,  Show  that  the  area 
of  the  parallelogram  formed  by  tangents  at  P,  P',  Q,  Q^  is 

Sa'^b 
(a  —  6)  sin  2<j>' 

94.  If  from  any  point  R  in*  the  plane  of  an  ellipse  the  perpendiculars 
RS,  RQ  are  drawn  on  the  equal  conjugate  diameters,  the  diagonal  PR  of 
the  parallelogram  PQRS  will  be  perpendicular  to  the  polar  of  R, 

95.  Normals  to  an  ellipse  are  drawn  at  the  extremities  of  a  chord  par- 
allel to  one  of  the  equi -conjugate  diameters;  prove  that  they  intersect  on 
a  diameter  perpendicular  to  the  other  equi -conjugate. 


262  THE  ELLIPSE   AND   HYPEBBOLA.  [157. 

96.  If  normals  are  drawn  at  the  ends  of  any  focal  chord  of  an  ellipse,  a 
line  through  their  intersection  parallel  to  the  major  axis  will  bisect  the 
chord. 

97.  If  circles  are  described  on  two  semi-conjug-ate  diameters  of  an 
ellipse  as  diameters,  the  locus  of  their  second  point  of  intersection  will  be 
the  curve 

2(x2 +  2/2)2  :z=a2x2  +  6V- 

98.  Show  that  the  angle  which  a  diameter  of  an  ellipse  subtends  at 
either  end  of  the  major  axis  is  supplementary  to  that  which  the  conjugate 
diameter  subtends  at  the  end  of  the  minor  axis. 

99.  If  6,  6^  are  the  angles  subtended  by  the  major  axis  of  an  ellipse  at 
the  extremities  of  a  pair  of  conjugate  diameters,  show  that  cot^  0  -f  cot^  d^ 
is  constant. 

100.  If  the  line  joining  the  foci  of  an  ellipse  subtends  angles  26 ^  20^  at 
the  ends  of  a  pair  of  conjugate  diameters,  show  that  tan^  6  -\-  tan^  6^  is  con- 
stant. 

101.  If  6,  O^are  the  angles  which  any  two  conjugate  diameters  subtend 
at  a  fixed  point  on  an  ellipse,  show  that  cot^  6  -|-  cot^  6^  is  constant. 

102.  The  normals  at  three  points  of  an  ellipse  whose  eccentric  angles 

are  a,  /3,  y,  will  meet  in  a  point  if 

sin  2a  sin  (iQ  —  7)  +  sin  2(3  sin  (7  —  a)  -f-  sin  27  sin  (a  —  /3)  =  0. 


CHAPTER  XI. 


POLAR  EQUATION  OP  A  OONIO,  THE  FOCUS  BEING  THE 

POLE. 

1 68.     To  find  the  polar  equation  of  a  conic,  the  focus  being  the  pole. 


M 

^ 

L 

^ 

X 

D 

F                N 

Let  F  be  the  focus,  DM  the  directrix,  and  e  the  eccentricity  of 
the  conic. 

Draw  FD  perpendicular  to  the  directrix,  and  let  DFX  be  taken 
for  the  positive  direction  of  the  initial  line. 

laet  L'FL  be  the  latus  rectum,  and  let 


l  =  FL=e'DF. 


(1) 


Let  P(/>,  0)  be  any  point  on  the  curve,  and  let  PM  and  PN  be 
perpendicular,  respectively,  to  DM  and  DX, 
Then  we  have 


FP=:e'MP=e'DN=e'DF-]-e'FN. 
Whence  p  =  l-\-  ep  cos  0. 


=  1  —  e  cos  0,     or    p  = 


e  cos  0 


(2) 
(3) 

(4) 


If  FD  is  taken  as  the  positive  direction  of  the  initial  line  and 
^  measured  clockwise,  the  equation  of  the  curve  is 


264  POLAR   EQUATION   OF   A   CONIC.  [168. 

-  zn  1  -f  e  COS  6'.  (5) 

If  the  axis  of  the  conic  makes  an  angle  y  with  the  initial  line, 
the  equation  of  the  curve  will  be  (§  68) 

-  z=  i_ecos(^  — r).  (6) 

If  the  conic  is  a  parabola,  we  have  e  =  1,  and  the  equation  may 
then  be  written 

Since  DF=—,  from  (1) ,  ^^e  equation  of  the  directrix,  DM,  is  (§  44) 
e 

I 

-  =  —  e  cos  0.  (8) 

The  equation  of  the  directrix  of  the  conic  (6)  is 

i:^_cos(^-r).  (9) 

The  perpendicular  distance  from  the  focus  (ae,  0)  to  the  asymp- 
tote whose  equation  is  6a?  —  a?/  =  0  is  [(5),  §  50 j 

abe       _  , 

Also  the  angle  which  this  asymptote  makes  with  the  principal 
axis  of  the  conic  is 

tan~^  -  =  cos  ^  —        ■■  =  cos  *  -. 

a  l/a'H-6'       .  e 

The  perpendicular  on  this  asymptote  from  the  focus  therefore 

makes  an  angle  — -f  cos~^— . 
z  e 

Hence,  by  §  44,  the  polar  equation  of  the  asymptote  is 

P  sin/ ^  —  cos-'- j  =  b.  (10) 

Similarly,  the  polar  equation  of  the  other  asymptote  is 

^  sin(^  +  cos-'-W  — 6.  (11) 


159.] 

1 59.     To  trace  the  curve  p  — 


POLAB  EQUATION  OF   A  CONIO. 
I 


265 


1  —  e  cos  d 

Since,  for  any  value  of  ^,  cos  0=  cos  ( —  ^),  the  curve  is  always 
symmetrical  about  the  initial  line  [§  29,  (1)]. 


I.     Let  e  =  1,  then  the  curve  is  a  parabola,  and  the  equation 
becomes  i 

P  — 


When  ^  =  0,  then  p 


1  —  cos  0 

00  .     As  ^  increases  continuously  from 


0  to  90°,  cos  0  decreases  from  1  to  0,  and  therefore  p  decreases 
continuously  from  infinity  to  I, 

As  d  increases  from  90°  to  180°,  cos  0  decreases  from  0  to  —  1, 
and  therefore  p  decreases  from  I  to  \l. 

Similarly,  as  0  varies  from  180°  to  270°,  p  increases  from  \l  to 
Z;  and  as  0  increases  beyond  270°,  p  continues  to  increase  until 
0  z=z  360°,  when  it  again  becomes  infinite. 

Hence,  for  0  increasing,  the  parabola  is  described  in  the  direc- 
tion 00  PLAL'P'  00  ,  as  shown  in  the  figure,  going  to  infinity  in 
the  direction  FX. 

It  is  to  be  noticed  that  p  is  always  positive,  not  changing  its  sign 
when  it  passes  through  infinity. 


266 


POLAR   EQUATION   OF   A   CONIC. 


[159. 


II.     Let  e  <  1,  then  the  curve  is  an  ellipse. 


At  the  point  A\  0  =  0  and  p  =  :j ,  which  is  positive,  since 

e<l. 

As  0  increases  from  0  to  90°,  cos  0  decreases  from  1  to  0,  and 


hence  p  decreases  from 
A'PBL. 


to  I.     We  thus  obtain  the  portion 


As  0  increases  from  90°  to  180°,  cos  0  decreases  from  0  to  —  1, 

and  therefore  p  decreases  from  I  to  - — ; — .     The  curve  therefore 

1  -f  e 

cuts  the  initial  line  again  at  some  point  A,  such  that  FA  =  — 


and  we  thus  obtain  the  portion  LA. 


l+«' 


Similarly,  as  0  continues  to  increase  from  180°  to  270°,  and 

then  to  360°,  p  increases  from :; — ; —  to  I,  and  then  to ,  and 

1  -\-  e        '  1  —  e 

we  have  the  portions  AL'  and  L'B'A'. 

Since  p  is  finite  for  all  values  of  0  when  e  <  1,  the  locus  is  a 
closed  curve  symmetrical  about  the  initial  line. 

III.     Let  e  >  1,  then  the  curve  is  a  hyperbola. 

When  ^  =  0,  1  —  e  cos  6*  =  1  —  e  =  —  (^  —  1). 

Hence  />  = -,  which  is  a  negative  quantity  since  e  >►  1. 

e  —  X 

The  corresponding  point  is  A'  in  the  figure. 


159.] 


POLAR  EQUATION  OF  A  CONIC. 


267 


.E' 


Let  6  increase  from  0  to  cos~^  —  =  a  =  XFKin  the  figure. 

Then  1  —  e  cos  0  increases  algebraically  from  —  (e  —  1)  to  —  0. 

,  Hence  p  decreases  algebraically  from to  — oo  . 

Therefore,  as  6  varies  from  0  to  a,  ^  is  negative  and  increases 
in  magnitude  from  FA'  to  infinity.  We  thus  obtain  the  portion 
of  the  curve  ^'5' Coo. 

If  6  is  very  slightly  greater  than  a,  then  cos  0  is  just  a  little 

less  than—,  so  that  1  — e  cos  0  is  very  small  and  positive,  and 
e 

therefore  p  is  very  great  and  positive. 

Hence,  as  0  increases  through  the  angle  a,  the  value  of  p  changes 
from  —  CO  to  +  00  . 

As  0  increases  from  a  to  tt,  1  —  e  cos  0  increases  from  0  to  1  +  e> 

and  therefore  p  decreases  from  «  to 


Now 


I 


1  +  e 


< 


1+e 
Hence  the  point  -4,  which  corresponds  to 


^  =  TT,  is  such  that  FA  <  FA', 


268  POLAB  EQUATION  OF  A  OONIO.  [159. 

As  6  increases  from  tt  to  27r  —  a  (the  reflex  angle  XFK'  in  the 
figure),  1  —  e  cos  6  decreases  from  1  +  e  to  0,  since 

cos  (27r  —  a)  =  cos  a  =  — 

so  that  p  increases  from  - — ^ —  to  oo  . 

1  +e 

Therefore,  corresponding  to  values  of  6  between  a  and  2?:  —  a, 
we  have  the  portion  of  the  curve  oo  CBADE  oo ,  for  which  p  is 


Finally,  as  6  increases  through  the  value  (2?:  —  a),  p  changes 
from  -j-  00  *o  —  00  ;  and  as  0  increases  from  2;r  —  a  to  27r, 
1  —  e  cos  0  decreases  algebraically  from  0  to  1  —  e,  and  therefore 

P  varies  continuously  from  —  co  to -.     Corresponding  to 

these  values  of  0  we  have  the  portion  of  the  curve  oo  E'D'A'-,  for 

which  p  is  negative. 

Thus  we  see  that  p  is  always  positive  for  the  right  branch  of 

the  curve,  and  negative  for  the  left  branch;  and  furthermore, 

that  as  6  varies  from  0  to  2?:,  the  complete  curve  is  described  in 

the  order 

A'B'  0'  00  00  CBADE  oo  oo  E'D'A'. 

,    The  lines  FK  and  FK'  are  parallel  to  the  asymptotes,  for 

1  a 


cos  a  =  cos  (2n  —  a) 


^        Va'  +  b' 


,',    tana  =  — ,     and    tan  (2?:  —  a)  = . 

Therefore  the  radius  vector  is  infinite  when  it  is  parallel  to 
either  of  the  asymptotes. 

If  a  line  FPQ  is  drawn  cutting  the  curve  in  two  points  P  and 
Q,  which  are  on  different  branches,  these  points  must  not  be  re- 
garded as  having  the  same  vectorial  angle.  The  radius  vector 
FQ  is  negative,  that  is  to  say,  FQ  is  drawn  in  the  direction  oppo- 
site  to  that  which  bounds  its  vectorial  angle ;  and,  therefore,  if  QF 
be  produced  to  H,  the  vectorial  angle  of  Q  is  XFH.  So  that,  if 
0  is  the  vectorial  angle  of  Q,  that  of  P  is  (<?  +  tt). 


160.]  POLAB   EQUATION   OF   A   CONIC.  269 

160.  To  find  the  polar  equation  of  the  straight  line  through  two 
given  points  on  a  conic,  and  to  find  the  polar  equation  of  the  tangent  at 
any  point. 

Let  F{pi,  a  —  /5)  and  Q(p2j  «  +  /5)  be  the  two  given  points. 
Let  the  equation  of  the  conic  be 

-  =  1  — ecos^.  (1) 

P 

If  A  and  B  are  arbitrary  parameters,  the  equation 

-  =  AcoaO  -{-B  cosine  — a)  (2) 

will  represent  a  straight  line ;  for  by  changing  to  rectangular  co- 
ordinates the  result  will  be  found  to  be  of  the  first  degree. 
Moreover,  this  line  can  be  made  to  pass  through  any  two  points, 
since  its  equation  contains  two  independent  parameters,  A  and  B. . 
It  will  pass  through  P  and  Q  if  the  values  of  A  and  B  are  so  de- 
termined that  p  will  have  the  same  value  in  (2)  as  in  (1)  when 
d  =  a  —  /?,  and  also  when  0  =  a  -\-  ^. 

Hence,  by  substituting  (a  —  /5)  and  (a  -{-  ^)  for  0  in  (1)  and 
(2),  we  have,  for  the  determination  of  the  proper  values  of  A 
and  B,  the  two  identities 


and 


Substituting  these  values  of  A  and  B  in  (2)  gives 

-  =  —  e  cos^  +  sec/3  co8(^  —  a),  (6) 

which  is  the  required  equation. 

If  j3  =  0,  the  two  points  P  and  Q  will  coincide  in  the  point 
whose  vectorial  angle  is  a,  and  the  line  (6)  will  become  the  tan- 
gent at  that  point.  Hence,  to  find  the  equation  of  the  tangent 
at  the  point  whose  vectorial  angle  is  a,  we  put  /3  =  0  in  (6),  and 
thus  obtain 

-=cos(<9  —  a) — ecos^.  (7) 


-  =  1  —  e  COS  (a  —  /9)  =  J.  cos  (a  —  /5)  +  J5  cos  /3, 

(3) 

-  =  1  —  e  cos  (a  H-  /5)  =  ^  cos  (a  +  /5)  +  J5  cos  /5. 

P2 

(4) 

.".     A=  —  e,     and     5  =  secy?. 

(5) 

270  POLAR  EQUATION  OF  A  CONIC.  [161. 

If  the  equation  of  the  conic  is  [§  158,  (6)] 

—  =  1  —  cos  ((9  —  Y)y 

the  equation  of  the  chord  joining  the  points  (a  —  /5)  and  (a  +  /^) 
will  be  (§  68) 

-  =  sec/5cos  (<?  —  a) — ecos(^  —  y))  (8) 

and  the  equation  of  the  tangent  at  the  point  a  will  be 

-  =  cos(^  —  a)  —  eGOS(<?  —  r).  (9) 

161.  To  find  the  polar  equation  of  the  polar  of  a  point  (p',  6')  with 
respect  to  the  conic 

-  =  1  — ecos^.  (1) 

P 

Let  P(iOi,  OL  —  ^)  and  Q^p^,  a  +  /?)  be  the  points  of  contact  of 
the  two  tangents  drawn  from  the  point  (/o',  0')  to  the  conic. 

Then  the  equation  of  the  line  PQ,  the  polar  of  (jo',  ^'),  in  terms 
of  a  and  /3  will  be  [(6),  §  160] 

-r=sec/5cos(^  —  a)  —  ecos^.  (2) 

The  values  of  a  and  /5  must  now  be  determined  in  terms  of  I,  e, 
p'y  and  0'. 

The  equations  of  the  tangents  at  P,  Q,  respectively,  are 

-  =  008(6  — a -\- IS)  — e  cos  0     [(7),  §160]      (3) 

—  =  cos((9  —  a  —  ^)  —  ecosO,  (4) 
Since  these  tangents  pass  through  (p'j  ^'),  we  have 

-  =  cos  (d'—  a  -j-  /f)  —  e  cos  ^'  (5) 

-  =:r:cos(^'— a  — /?)  — ecos^'.  (6) 


161.]  POLAR  EQUATION   OF   A   CONIC.  271 

Subtracting  (6)  from  (5)  gives 

cos  {0'—  a  -{-  /?)r=  COS  {6'—  a  —  /?).  (7) 

.-.     tf'_a_^y5=:_(6''_a  —  /3),  since /5  9^0.  (8) 

.-.      a  =  0'.  (9) 

From  (9)  and  (5)  we  now  get 

cos  /?  =  -  +  e  COS  e\  (10) 

Substitute  these  values  of  a  and  cos  /3  in  (2),  and  we  have 

(-  +  e  cos  o\l-,  -}-  6  cos  o\  =  cos  (^  —  0'),  (11) 

which  IS  the  required  equation. 

Examples  on  Chapter  XI. 

1.  In  a  parabola,  proye  that  the  length  of  a  focal  chord  which  is  inclined 
at  30°  to  the  axis  is  four  times  the  length  of  the  latus  rectum. 

2.  In  any  conic  section  the  semi-latus  rectum  is  a  harmonic  mean  be- 
tween the  segments  of  any  focal  chord.    (See  note  under  §  115.) 

3.  The  sum  of  the  reciprocals  of  two  perpendicular  focal  chords  of  any 
conic  is  constant. 

4.  If  PFP^  and  QFQ^  are  any  two  focal  chords  of  a  conic  at  right  angles 
to  one  another,  show  that  p^ryp?  +  qf-fQ'  ^^  ^o^^tant.* 

5.  The  tangents  at  two  points  P  and  Q  of  a  conic  meet  in  T.  If  F  is  the 
pole,  show  that  the  vectorial  angle  of  T  is  half  the  sum  of  the  vectorial 
angles  of  P  and  Q. 

Hence,  prove  that  if  P  and  Q  are  on  different  branches  of  a  hyperbola, 
then  FT  bisects  the  supplement  of  the  angle  PFQ,  while  in  all  other  cases 
FT  bisects  the  angle  PFQ, 

6.  If  the  conic  is  a  parabola  and  a,  /?  are  the  vectorial  angles  of  P,  Q, 
show  that  the  coordinates  of  T  are  given  by  the  equations 

^-K«  +  /3),        ^  =  2sin|sin|: 

7.  If  the  conic  is  a  parabola,  then 

FT'  =  FPFQ. 

*  In  this  list  of  examples  F  will  always  be  the  focus. 


272  POLAR   EQUATION   OF   A   CONIC.  [161. 

8.  If  the  conic  is  central  and  b  is  the  semi-minor  axis,  then 

1 L_-l 

PF'FQ       FT'^V 

9.  If  a  tangent  at  any  point  P  of  a  conic  meets  the  directrix  in  IT,  then 
the  angle  KFP  is  a  right  angle.    (O/.  Ex.  39,  p.  178.) 

10.  If  the  tangents  at  the  two  points  P,  Q  of  a  conic  meet  in  T,  and  if 
the  straight  line  PQ  meets  the  directrix  corresponding  to  F  in  K^  then  the 
angle  KFT  is  a  right  angle. 

11.  Prove  that  perpendicular  focal  chords  of  a  rectangular  hyperbola 
are  equal. 

12.  If  a  straight  line  drawn  through  the  focus  of  a  hyperbola,  parallel 
to  an  asymptote,  meets  the  curve  in  P,  prove  that  FP  is  one-fourth  of  the 
latus  rectum. 

13.  Show  that  the  length  of  any  focal  chord  of  a  conic  is  a  third  pro- 
portional to  the  transverse  axis  and  the  diameter  parallel  to  the  chord. 

14.  If  chords  of  a  conic  subtend  a  constant  angle  2/3  at  the  focus,  the 
tangents  at  the  ends  of  the  chords  will  meet  on  a  fixed  conic  whose  focus 
is  P,  and  the  chords  will  touch  another  conic  whose  focus  is  F.  Consider 
the  cases  in  which  cos/3  >,   =,  and   <  e. 

15.  The  exterior  angle  between  any  two  tangents  to  a  parabola  is  equal 
to  half  the  difference  of  the  vectorial  angles  of  the  points  of  contact. 

16.  The  locus  of  the  point  of  intersection  of  two  tangents  to  a  parabola 
which  intersect  at  a  constant  angle  is  a  hyperbola  having  the  same  focus 
and  directrix  as  the  given  parabola.    (Exs.  14  and  15.) 

17.  If  PQ  and  PR  are  two  chords  of  a  conic  subtending  equal  angles  at 
the  focus,  the  tangent  at  P  and  the  chord  Q,R  will  intersect  on  the  directrix. 
Hence,  if  the  sum  of  the  vectorial  angles  of  two  points  is  constant,  the 
chord  through  those  points  will  meet  the  directrix  in  a  fixed  point. 

18.  PQ,  is  a  chord  of  a  conic  which  subtends  a  right  angle  at  a  focus. 
Show  that  the  locus  of  the  pole  of  PQ  and  the  locus  enveloped  by  PQ  are 
each  conies  whose  latera  recta  are  to  that  of  the  original  conic  as  l/2:l 
and  1 :  l/2,  respectively. 

19.  PFQ  is  a  focal  chord  of  a  conic,  and  a  parallel  chord  AP^  through 
the  vertex  A  meets  the  latus  rectum  in  Q^.  Prove  that  the  ratio 
PF  •  FQ  :  AP^  •  ^Q^  is  constant. 

20.  If  ^,  P,  C  are  any  three  points  on  a  parabola,  and  the  tangents  at 
these  points  form  a  triangle  A'B'C^y  show  that  FA'FBFC= FA'  •  FB'  •  FG\ 


161.]  POLAR   EQUATION   OF   A   CONIC.  273 

21.  Show  that  the  equations  —  =  ±:  1  —  e  cos  6  represent  the  same  conic. 

If  (p,  0)  is  a  point  on  the  curve  when  the  upper  sign  is  taken,  what  will  be 
its  coordinates  when  the  lower  sign  is  used  ? 

22.  The  product  of  the  segments  of  any  focal  chord  of  any  conic  is  equal 
to  the  product  of  one-fourth  the  latus  rectum  and  the  whole  chord. 

23.  Show  that  the  polar  equation  of  the  normal  at  the  point  whose  vec- 
torial angle  is  a  is 

-  =  e  sin  6  —  sin  (^  —  a). 


1  —  e  cos  a    p 

24.  If  a  focal  chord  of  an  ellipse  makes  an  angle  a  with  the  axis,  the 
angle  between  the  tangents  at  its  extremities  is 

,  2e  sin  a 
tan-^  ~. -^. 

25.  By  means  of  the  equation  —  =  1  —  e  cos  ^,  show  that  the  ellipse 

might  be  generated  by  a  point  which  moves  so  that  the  sum  of  its  distances 
from  two  fixed  points  is  constant. 

26.  If  a  chord  of  a  conic  subtends  a  constant  angle  2/3  at  the  focus,  prove 
that  the  locus  of  the  point  where  it  intersects  the  internal  bisector  of  the 
angle  2/3  is  the  conic  (c/.  Ex.  14) 

i  cos  /3  .  no 

=  1  —  e  cos  /?  cos  6. 

P 

27.  Show  that  the  locus  of  the  middle  points  of  focal  chords  of  a  conic 
section  is  a  conic  whose  equation  is 

_     le  cos  6 
^"l  —  e^cos^r 

Show  that  in  Cartesian  coordinates  this  equation  becomes 

and  discuss  this  result  for  different  values  of  e. 

28.  Given  the  focus  and  directrix  of  a  conic,  show  that  the  polar  of  a 
given  point  with  respect  to  it  passes  through  a  fixed  point. 

29.  Two  conies  have  a  com.mon  focus.    Prove  that  two  of  their  common 

chords  pass  through  the  intersection  of  their  directrices. 

I  V 

Sug.    If  the  conies  are  —  =  1  —  e  cos  d  and  —  =  1  —  e^  cos  {d  —  7),  the 

I  r  V  ~\ 

common  chords  are  — [-e  cos  ^  =  i     — \-e^  cos  {d  —  y)   . 

30.  P  is  any  point  on  a  conic  and  a  straight  line  is  drawn  through  F  at 
a  given  angle  with  FP  to  meet  the  tangent  at  P  in  T.  Prove  that  the  locus 
of  Pis  a  conic  whose  focus  and  directrix  are  the  same  as  those  of  the 
original  conic. 

19 


274  POLAR   EQUATION   OF   A   CONIC.  [161. 

31.  A  circle  whose  diameter  is  d  passes  through  the  focus  F  of  a  conic 
and  meets  it  in  four  points  whosfe  distances  from  F  are  pi,  p2,  Pa,  and  pi» 
Prove  that 

(1)     PUhPiPi  =  -^\ 

and  (2)    -  H h  -  =  t- 

Pi       P2      Pz      Pi       I 

32.  A  circle  whose  centre  is  at  a  fixed  point  on  the  principal  axis  inter- 
sects the  conic  in  four  points.  Prove  that  the  sum  of  their  distances  from 
the  focus  is  constant. 

33.  If  PFQ,  and  BF^R  are  two  chords  of  an  ellipse  through  the  foci  F  and 

PF       PF^ 
F^f  then  will  ^^^  +  'Wp  ^®  independent  of  the  position  of  P. 

34.  Two  conies  have  the  same  focus,  and  the  distance  of  this  focus  from 
the  corresponding  directrix  of  each  is  the  same ;  if  the  conies  touch  one 
another,  prove  that  twice  the  sine  of  half  the  angle  between  their  principal 
axes  is  equal  to  the  difference  of  the  reciprocals  of  their  eccentricities. 

[Write  the  equations  of  the  common  tangent  at  /5,  compare  the  coeffic- 
ients of  sin  0  and  cos  ^,  and  eliminate  ^.] 

35.  Show  that  the  equation  of  the  circle  circumscribing  the  triangle 
formed  by  three  tangents  to  the  parabola  —  =  1  —  cos  0  drawn  at  points 
whose  vectorial  angles  are  a,  ^5,  and  y  is 


^  ..^  ^  ^„«  (^  ^„^  y  .,•«  /'^  +  /5  -h  y 


p  =  j^  CSC  s-  CSC  ^  CSC  ^  sm 


.); 


2        2        2        2        V        2 
and  therefore  it  always  passes  through  the  focus.    (See  Ex.  6.) 

38.  A  comet  is  moving  in  a  parabolic  orbit  around  the  sun  at  its  focus 
and  when  at  100,000,000  miles  from  the  sun,  the  radius  vector  makps  an 
angle  of  60°  with  the  axis  of  the  orbit.  What  is  the  polar  equation  of  the 
comet's  orbit?    How  near  does  it  approach  to  the  sun  ? 

.  50,000,000 

^    Ans.    p  =  -r^ '—^  . 

1  —  cos  0 

37.  Two  straight  lines  bisect  each  other  at  right  angles.  Prove  that  the 
locus  of  the  points  at  which  they  subtend  equal  angles  is 

p"^    _a  cos  6  —  5  sin  ^ 
ah  ~  h  cos  ^  —  a  sin  0^ . 

2a  and  25  being  the  lengths  of  the  lines,  their  point  of  intersection  the  pole. 
Is  the  locus  a  conic? 


CHAPTEK  XII. 
THE  GENERAL  EQUATION  OF  THE  SECOND  DEGREE. 

Construction  of  Curves  of  the  Second  Order. 

1 62.  The  most  general  equation  of  the  second  degree  in  Car- 
tesian coordinates  is  of  the  form 

ax"  +  '^hxy  +  hy'  J^^gx  ■^2fy  +  0=^0.  (1) 

From  the  investigations  of  §  108  and  §  67  it  follows  that  this 
equation  always  represents  a  conic  section,  whether  the  axes  are 
rectangular  or  oblique.  In  order  to  determine  the  general  nature 
of  this  conic,  and  to  draw  the  curve  in  its  proper  position  with 
1  eference  to  the  axes  of  coordinates,  solve  the  equation  with  re- 
spect to  one  of  the  variables,  y  say. 

There  are  two  cases  to  be  considered,  according  as  y  appears  in 
the  equation  to  the  second,  or  only  to  the  first  degree ;  i.  e.  ac- 
cording as  6  :?^  0,  or  6  :=  0. 

First  suppose  that  6  7^  0,  and  solve  (1)  with  respect  to  ?/;  we 
thus  obtain 

hx^fl 


y-= ^^lV(ih'—ab)x'  +  2(ifh  —  bg)x-h(f—be),     (2) 

hx+f      1 


or  y  = ^±^\/Lx'-\-2Mx-^N,  (3) 

where  L  =  h^ —  ah,     M=fh  —  bg,     N=f  —  be. 

Thus  for  any  given  value  of  x  there  are  two  values  of  y. 

Let  ,y=-^J^,  (4) 

and  Y=  ^VLx'  +  2Mx  +  N.  (5) 

Then  equation  (3)  takes  the  form 

y=if±Y..  ,    (6) 


276 


GENERAL  EQUATION  OF  THE  SECOND  DEGREE. 


[162. 


Draw  the  line  BiE"  represented  by  equation  (4). 

Then  equation  (6)  shows  that  for  any  given  value  of  Xj  say  OQy 
the  ordinates  of  the  curve,  QP,  QF',  may  be  found  by  adding  to 
and  subtracting  from  the  corresponding  ordinate,  ?/'==  QR,  of  the 
line  HK,  the  same  quantity  Y, 

.-.     P'R  =  RP=Y,     and     P'P  =  2Y,  (7) 

Hence  the  chord  joining  two  points  of  the  curve  which  have 
the  same  abscissa,  i.  e.  any  chord  parallel  to  the  i/-axis,  is  bi- 
sected by  the  line  HK.  This  line  is,  therefore,  a  diameter  of  the 
curve  (§126),  and  Fis  the  length  of  the  ordinate  measured  from 
this  diameter.  The  form  of  the  curve,  therefore,  is  determined 
by  the  nature  of  the  function  F;  and  the  construction  of  the 
locus  is  reduced  to  the  study  of  the  trinomial 

Lx"  +  "IMx  +  N. 

Let  x'  and  x"  be  the  roots  of  this  trinomial,  x'  being  the  smaller ; 
then  we  may  write  (§89) 

1 


F=- 


VL{x  —  x'){x—x"). 


(8) 


Draw  the  two  lines  Q'A'  and  Q"A",  whose  equations  are,  re- 
spectively, 

x=^x'     and     x  =  x" . 


163.]      GENERAL  EQUATION  OF  THE  SECOND  DEGREE.       277 

Now  when  x=^x'==OQ',  and  also  when  x  =  x"^^OQ"  in  equa- 
tion (3),  and  hence  in  (8),  we  have  F=0. 

.-.     P'P=0,   from  (7). 

That  is,  the  curve  intersects  each  of  the  lines  Q'A',  Q^'A"  in 
two  coincident  points  at  A',  A"^  respectively.  Therefore  the 
lines  Q'A'  and  Q"A"  are  tangents  to  the  curve ;  and  since  they 
are  parallel  to  the  chords  bisected  by  HK,  the  points  of  contact 
are  the  extremities  of  the  diameter  HK.     (§  148,  Cor.  IV. ) 

As  the  form  of  the  conic  depends  mainly  on  the  sign  of  the  co- 
eflBicient  L,  there  are  three  principal  cases  to  be  considered,  each 
of  which  may  be  subdivided  into  several  others,  according  to  the 
nature  of  th6  roots  a?',  x"  of  the  trinomial. 


The  Ellipse.     L~h^  —  aft  <  0. 

1 63.    Consider  the  case  when  the  coefficient  L  has  a  negative 
value,  say  —  K.     We  may  then  write  [(8),  §  162] 

Y=l\/—K{x  —  x'){x  —  xny 


OT^  =^\/K{x'—x){x  —  x"). 

I.     Let  M^—  LN=  M^  +  KNy>  0. 

Then  x'  and  x"  are  real  and  unequal. 

When  X  is  either  less  than  x',  or  greater  than  a;",  the  two 
factors  (ic' — x)  and  {x  —  x")  have  different  signs;  hence  their 
product  is  negative,  and  therefore  Y  is  imaginary.  That  is, 
there  are  no  real  points  on  the  locus  to  the  left  of  Q'A'  or  to  the 
right  of  q'A".     (Fig.  §  162. ) 

For  every  value  of  x  taken  between  the  limits  oc^  and  x"  (i.  e, 
such  that  a?'<a?<a?")  the  factors  (x' — x)  and  (x — x")  are 
both  negative ;  hence  their  product  is  positive,  and  therefore  Y  is 
real.  Moreover,  as  x  varies  from  x'  to  x"j  F  starts  with  the  value 
zero,  is  always  finite,  and  returns  to  zero.  The  locus  is,  therefore, 
a  closed  curve  lying  between  the  two  tangent  lines  Q'A'  and  Q"A", 
and  passing  through  the  points  A',  A", 

Therefore  the  conic  is  an  ellipse. 


278 


GEXERAL  EQUATION  OF  THE  SECOND  DEGREE. 


[163. 


II.     Jjet  M'—LN  =  M'-\-KN=0. 

Then  the  two  roots  x'j  x"  are  equal,  and  we  have 


r=^(. 


^IV^tk. 


The  quantity  F  is,  therefore,  imaginary  for  all  values  of  a?,  ex- 
cept X  =  a:',  and  then  F  =  0.  The  two  tangents  §' J.  and  i^^'A" 
then  coincide,  A"  coincides  with  A' ^  the  locus  reduces  to  a  single 
point  on  the  diameter  HK.,  an^  is  called  a,  point  ellipse. 


III.     Let  3P  —LN=M'^  KN  <  0. 
The  trinomial  may  be  written 


—  Kx'-\-2Mx-\-N= 


4("^--^)— T^j- 


M'  +  KN^ 


The  quantity  within  the  bracket  is  positive  for  all  values  of  x^ 
since  M^  +  ^^  is  negative.  Hence  F  is  imaginary  for  every 
value  of  X.  The  given  equation,  therefore,  has  no  real  solution, 
and  consequently  does  not  represent  a  real  geometrical  locus. 


Ex.    Trace  the  conic  Sx^  —  2xy  -\- 


Here  L  =  7i2_  a6  =  1  —  6  =  —  5. 

Hence  the  curve  is  an  ellipse.    Solving  the  equation  with  respect  to  y 
gives  

y  =  ijc  +  2  ±  ^V—hxix  —  '^). 


164.]  GENERAL  EQUATION   OF   THE  SECOND   DEGREE.  279 

Therefore  we  have  the  diameter  HK  represented  by  the  equation 

and  F  =  J  v'— 5a;(a;  — 8). 

Placing  the  quantity  under  the  radical  sign  equal  to  zero  gives 

5x(a;  — 8)=0. 

The  roots  of  this  equation  are  a;^=  0  and  x^^=  8. 

When  X  is  less  than  0,  or  greater  than  8,  Y  is  imaginary ;  while  for  all 
values  of  x  between  0  and  8,  Fis  real  and  finite,  but  reduces  to  0  both  when 
X  =  0  and  when  x  =  %. 

Therefore  the  curve  lies  between  the  tangents  a;  =  0  and  a?  =  8. 

The  equation  of  the  diameter  BB^  parallel  to  the  2/-axis,  and  therefore 
conjugate  to  HK  (§  149),  is 

a;  =  J(x'-fx^0  =  4. 

When  ic  =  4,  then  F  =  2/5  =  4.4+. 

Measure  CB  =  CB^=  4.4,  and  through  B,  B^  draw  lines  parallel  to  HK. 
The  curve  lies  within  the  parallelogram  DEFG  thus  formed,  and  is  tangent 
to  its  sides  at  the  points  A,  A\  B,  B^. 

The  Parabola.     L  =  h^  —  ah  =  0. 

164.  Suppose  next  that  the  coefficient  L  is  zero.  Equation 
(5),  §  162,  then  takes  the  form 

One  root  x"  of  the  trinomial  is  now  infinite  (§  98,  III.),  and 
consequently  the  tangent  Q"A"  has  moved  off  to  an  infinite  dis- 
tance. 

I.     If  M  ^  0,  the  equation  of  the  other  tangent  Q'A'  is 

_       ^ 
^~      23f 

N 
As  X  varies  from  —  ^^  to  infinity  on  the  positive  side  of  this 

line,  the  values  of  Y  are  real  and  vary  from  0  to  oo  ;  while  for  all 
values  of  x  on  the  negative  side  of  this  tangent  F  will  be  imaginary. 
The  conic  is  therefore  a  parabola,  since  it  consists  of  a  single  in- 
finite branch.  It  passes  through  the  point  A\  lies  on  the  positive 
side  of  the  tangent  2Mx  +  iV  =  0,  and  HK  is  the  diameter  which 
bisects  all  chords  parallel  to  the  y-a.xis. 


280 


GENERAL  EQUATION  OF  THE  SECOND  DEGREE. 


[164. 


II.  If  M=  0,  both  tangents  are  at  infinity  (§  98,  III.),  and 
the  given  equation  (3),  §  162,  reduces  to 

hx^f      1    ,^^ 

y  = i-^iV^' 

If  N  is  positive,  this  equation  represents  two  real  straight  lines 
which  are  parallel  to  the  diameter  HK  and  equidistant  from  it. 
If  J\r  =  0,  these  two  parallel  lines  coincide  with  HK.  If  iV  is 
negative,  the  equation  has  no  real  solution,  and  therefore  the  two 
lines  are  imaginary. 

Ex.    Let  the  given  equation  he 

x2  +  4x2/ +  4i/2  +  8x  —  81/ —  32  =  0. 
Solving  for  y  we  have 

2/  =  —  ^x  +  1  ±  i>^— 12xH-36. 
.  • .    1/  —  0,  and  the  curve  is  a  parabola. 


The  equation  of  HK  is 


y  =  —  lx  +  i, 


and 


F=^i/— 12X-1-36. 


When  a;  =  3,  F  =  0;  and  when  x  >  3,  F  is  imaginary. 
Therefore  the  curve  lies  to  the  left  of  the  line  x  =  3. 
When  X  =  0, 2/  =  1  ±  3.    Hence  the  curve  passes  through  the  points 
(0,4)and(0,  —  2). 


165.] 


GENERAL  EQUATION  OF  THE  SECOND  DEGREE. 


281 


The  Hyperbola.     L  =  h^  —  a6  >  0. 
165.     Finally,  suppose  that  L  is  positive. 
Then  Lx'  +  2Mx  +  ^=  Mx  —  x')(x  —  x"), 


and 


y=^VL(x  —  s(f)ix  —  x"). 


I.     'LetM'—LN>0. 

Then  x'  and  a::'' are  real  and  unequal. 

When  X  is  greater  than  of  and  less  than  a?"  ({,  e,  af<.x<i  x")y 
Y  is  imaginary.  That  is,  there  are  no  points  on  the  curve  lying 
between  the  two  tangents  Q'A'  and  Q"A".  As  x  varies  from 
ic"  to  -f-  00  ,  or  from  a?'  to  —  x  ,  Y  is  real  and  varies  from  0  to  oo . 
Hence  the  curve  consists  of  two  distinct  infinite  branches,  the 
one  lying  to  the  right  of  Q"A",  the  other  to  the  left  of  QA'. 
That  is,  the  conic  is  a  hyperbola. 


282  GENERAL    EQUATION    OP    THE    SECOND    DEGREE.  [165. 

The  Asymptotes. 

Since      Lx'  +  2Mx  -^N-L\x^^\—         ^        , 

the  given  equation  [(3),  §162]  may  be  written 


hx-^f      1     L/      ,    MV     M'  —  LN 


Consider  now  the  equation 


um 


hx-{-f      1     k/      ,    My 


When  X  has  a  very  large  numerical  value,  the  first  term  under 
the  radical  sign  in  (1)  is  very  large  as  compared  with  the  numeri- 
cal value  of  the  second  term.  Hence,  as  x  approaches  either 
+  00  or  —  00 ,  the  limiting  values  of  y  in  (1)  are  the  corre- 
sponding values  of  y^  given  by  (2).  That  is,  the  conic  and  the 
two  straight  lines  BE  and  B'E'  represented  by  (2)  come  together 
at  infinity.      (C/.  §116.) 

Therefore  (2)  is  the  equation  of  the  as3^mptotes,  and  may  be 
written 

(hx^f)  _^1/      ,    M\   /^ 


y=       h 


\{-^^yL.      (3) 


II.  Letir  — i.iV^=0. 

The  given  equation  (1)  then  takes  the  same  form  as  (2),  and 
therefore  represents  two  straight  lines. 
The  roots  of  the  trinomial  are  then 

Hence  the  two  lines  intersect  on  the  diameter  HK  in  the  point 

M 

for  which  x  = ^,  since  this  value  of  x  makes  F=  0. 

III.  Letir  — Xi\r<o. 

Then  the  two  roots  ocf  and  x"  of  the  trinomial  are  imaginary. 
In  this  case  the  curve  has  no  tangents  parallel  to  the  ?/-axis. 
The  trinomial 

Lx^  -\-2Mx  -\-  N  =  Lyx -\- -y)  H J 


165.]     GENERAL  EQUATION  OF  THE  SECOND  DEGREE.       28o 

is  now  the  sum  of  two  positive  quantities,  and  therefore  the  value 
of  Y  is  real  for  all  values  of  x  and  never  becomes  zero.  Hence  the 
curve  does  not  cut  the  diameter  HK.  The  given  equation  may 
now  be  written 


y  =  --^±\^L[.  +  ^)+t^^^r^,         (4) 


J£+/..lJrY..^£V4--^^-^^' 


which  shows  that  the  asymptotes  are  tlie  two  lines  given  by 
equation  (3),  as  under  condition  I. 

Comparing  equations  (1),  (2),  and  (4),  we  see  that  for  the  same 
value  of  X  the  value  of  Fis  least  in  (1)  and  greatest  in  (4);  i.  e. 
RP  <  RP,  <  RP^.  Therefore  the  loci  of  (1)  and  (4)  lie  in  differ- 
ent angles  of  the  asymptotes. 

The  value  of  y  given  hy  (4)  is  least  when  the  first  term  under 
the  radical  sign  is  zero ;  i.  e.  when 

Since  this  line  (5),  H'K'  in  the  figure,  passes  through  the 
intersection  of  the  asymptotes,  and  is  parallel  to  the  ^/-axis,  it  is 
the  diameter  conjugate  to  HK,  and  therefore  bisects  all  chords 
parallel  to  IT^  (§  149). 

It  should  be  noticed  that  if  the  change  in  the  sign  of  if^  —  LN 
is  due  to  a  change  in  the  value  of  the  constant  term  c  of  the 
given  equation  (which  cannot  affect  L  and  M,  §  162),  the  two 
conies  represented  by  (1)  and  (4)  will  have  the  same  asymptotes. 
(C/.  §116,  XL,  and  §146.) 

If  M"^  —  LN  <  0,  it  is  generally  more  convenient  to  solve  the 
given  equation  with  respect  to  x. 

Ex.     Trace  the  curve 

43/2  —  Ixy  —  2x2  —  4a;  —  7i/  +  jgi  ^  q.  (1) 

Solving  for  y  gives 


2,  =  ^J7a;  +  7±9/a:2  +  2x-3J.  (2) 

Since  L  is  now  positive,  the  curve  is  a  hyperbola. 

The  equation  of  HK  is 

y  =  Kx-\-iy, 


and  F  =  |v/(x  +  3)(x  — 1). 

.-.    x^=  — 3,    x''=U 


284       GENERAL  EQUATION  OF  THE  SECOND  DEGREE.      [166. 

Y 


Hence  the  equations  of  the  tangents  Q^A^  and  Q^^A^^  are 

a; +  3  =  0    and    x  —  1  =  0.  (3) 

When  X  takes  any  value  between  the  limits  —  3  and  + 1,  the  value  of  Y 
is  imaginary,  so  that  no  part  of  the  curve  lies  between  the  lines  a;  +  3  =  0 
and  x  =  i. 

Completing  the  square  under  the  radical  sign  in  equation  (2),  we  have 

2/=r^J7a;  +  7±9/(x  +  l)2  — 4j.  (4) 

Dropping  the  constant  —  4,  we  finally  have 

2/  =  Ka5  +  l)±|(x  +  l).  (5) 

Therefore  the  equations  of  the  asymptotes  BE,  B^E^  are 

2/  =  2x  +  2    and    41/  +  x  -I-  1  =  0.  (6) 

The  curve  is  the  hyperbola  shown  in  the  figure. 

1 Q6,  6=0.  It  has  so  far  been  assumed  that  the  coefl&cient 
b  was  not  zero.  If  6  =  0,  and  a^O,  the  equation  can  be  solved 
with  respect  to  x  and  the  curve  can  be  traced  by  the  methods 
already  given.  When  either  a  or  6  is  zero,  the  conic  is  a  hyper- 
bola, for  h?  —  ah  is  then  positive,  unless  h  is  also  zero,  when  the 


166.] 


GENERAL  EQUATION  OF  THE  SECOND  DEGBEE. 


285 


locus  is  a  parabola.  In  case,  however,  a  variable  appears  only 
in  the  first  power,  it  is  preferable  to  solve  the  equation  with 
respect  to  that  variable. 

Suppose  the  given  equation  to  be 


ao^  +  2hxy  +  '^gx  +  2/i/  +  c  =  0. 


Solving  with  respect  to  y  gives 


y 


-  2(^0:+/)    • 


(1) 


(2) 


If  we  perform  the  division  in  the  second  member  of  (2)  until 
a  remainder  is  found  that  does  not  contain  aj,  the  equation  wiU 
reduce  to  the  form 


y  =  a'x^-h'-\- 


X  —  d' 


(3) 


Let  AB  be  the  line 
y  =  a'x  +  6', 
and  DE  the  line 


x  —  d  =  ^. 


Let 


Y= 


d 


=  RP, 


and  suppose  c'  >  0,  in  order  to 
fix  the  ideas. 

For  any  given  value  of  a;,  say  OQ,  there  is  only  one  value  of  F, 
and  only  one  value  of  y  given  by  (1),  which  is  found  by  adding 
the  quantity  Y=  RP  to  the  ordinate  QR  of  the  line  AB.  More- 
over, Y  is  positive  or  negative  according  as  o^  >  or  <  c?  (since 
c'  >  0) ;  i.  e.  according  as  QP  is  to  the  right  or  left  of  ED.  When 
X  is  very  slightly  less  than  d,  Y  is  very  great  and  negative; 
when  X  is  very  slightly  greater  than  c?,  Y  is  very  great  and 
positive.  As  cc,  increasing,  approaches  d,  i.  e.  as  QR  approaches 
ED  from  the  left,  F=  RP  =  —  go  ;  while  as  QR  approaches  ED 
from  the  right,  x  decreasing  to  d,  Y==RP=  -f  oo  .  Further- 
more, when  a?  =  -|-  00  or  —  go  ,  F=  0. 

Therefore  the  two  lines  AB  and  ED  are  the  asymptotes,  and 
the  curve  lies  in  the  angles  ACE  and  BCD, 


286 


GENERAL  EQUATION  OF  THE  SECOND  DEGREE. 


[166. 


If  d  <  0,  then  Y  will  be  positive  when  x  <id^  and  negative 
when  x^  d.  Hence  the  curve  will  lie  in  the  angles  A  CD  and 
BCE. 

If  c'=  0,  the  numerator  of  the  second  member  of  (2)  is  divis- 
ible by  the  denominator,  and  the  given  equation  may  be  written 

{y  —  a'x  —  b')(x  —  d)=0, 

which  represents  two  intersecting  lines,  one  of  which  is  parallel 
to  the  2/-axis. 

It  a  =  b  =  0,  the  conic  can  be  traced  in  the  same  way.  We 
then  have  a'  =  0,  and  therefore  the  asymptotes  are  parallel  to  the 
axes  of  coordinates. 

If  b  and  h  are  both  zero,  the  general  equation  takes  the  form 
y  =  a! 31?  A-  b'x  +  c'.  The  curve  is  then  a  parabola  with  axis  par- 
allel to  the  2/-axis,  and  is  easily  constructed. 

Ex .     Trace  the  curve  x^  +  ^xy -\-^ -\-x-\-^—^* 

D         Y 


Solving  for  y  gives 


3Vx  +  2/- 


When    a;  <  —  2,     F  is  positive. 
When    (c  >  —  2,     F  is  negative. 

X  — 1 


When    X  —  —  2,     Y"  is  infinite. 
When    x  =  +QO,or  —  oo,     F  =  0. 

and    a;+2  =  0 
6 

are  the  equations  of  the  asymptotes  AB  and  ET>. 

Tho  curve  passes  through  the  points  P(0,  —  1)  and  P\ —  3,  4) ;  its  posi- 
tion with  reference  to  the  axes  is  shown  in  the  figure. 


166.]  GENERAL   EQUATION   OP   THE   SECOND   DEGREE.  287 

EXAMPLES. 

Trace  the  conies  given  by  the  following  equations : 

1.  x^  —  ixy-\-iy''  —  8y  =  0. 

2.  5a;*  +  4x3/  +  42/2_12x  — 242/  =  0. 

3.  2x2  _|.  2xy  _  42/2  —  6x  +  61/  ±  9  =  0. 

4.  10x2  +  6a;2/  +  2/2  +  8a;  +  4y  +  8  =  0. 

5.  4x2— 12x3/+9y2  +  6x  — 9t/  — 4  =  0. 

6.  4x2  — 6x2/  — 42/2 +  28x4- 4?/ +  49  =  0. 

7.  4x2  —  4x2/ +  2/' 4- 12x  +  62/ +  25  =  0. 

8.  16x2  _  iQxy  ^  122/2  +  96x  —  582/  +  81  =  0: 

9.  13x2  +  6x2/  +  2/'  +  4  =  0. 

10.  X2/  +  3X— 22/  =  0. 

11.  45x2/  — 85x  — 632/ +  119  =  0. 

12.  4x2/— x2  +  4x-82/  — 8dr24  =  0. 

13.  x2  +  4x2/  +  42/'  -h  4x  +  82/  —  12  =  0. 

14.  2/'  — 2x2/  + 2x  + 2/  — 2  ±2  =  0. 

15.  3x2  — 4x2/  +  22/2+2x-82/  +  17  =  0. 

16.  9x2  — 24x2/  + 162/2  +  48X- 642/ +  64  =  0. 
\              17.  2x''  +  7x2/~42/2  +  4x  +  72/  +  22i  =  0. 

18.  4x2  — 4x2/  + 2/'  + 4x  — 22/ +  5  =  0. 

19.  x*  +  3x2/  +  9x  +  32/  +  18±18  =  0. 

20.  9x2  +  12x2/  +  42/2  — 7x  — 82/  — 11=0. 

21.  6x2  +  7x2/  — 32/2  +  5X+132/  — 4  =  0. 

22.  3x2  — 2x2/ +  22/2  — 22x  — 62/ +  27  =  0. 

23.  4x2  +  7x2/  — 22/2+ 15x  + 32/ +  90  =  0. 

24.  2x2— 2x2/  +  2/'  — 2x  +  2y+3  =  0. 

25.  16x2  +  24x2/  +  92/'  -  16x  —  122/  +  6  =  0. 

26.  2^2_^2x2/  +  5x  — y  — 20  =  0. 

27.  12x2/  — 33x  + 82/  — 66  =  0. 

28.  91x2— 117x2/  — 21x  + 272/  — 33  =  0. 


288  GENERAL   EQUATION   OF   THE   SECOND  DEGREE.  [167. 


To   Transform  the   General   Equation  of  the  Second   De- 
gree TO  One  of  the  Standard  Forms. 

167.     Whenh'—abz^O. 

It  has  been  shown  in  §  109  that  the  equations  for  finding  the 
centre  (a?',  y')  of  the  conic  represented  by  the  general  equation 

ax'  +  2hxy  +  by'  -^  2gx -{- 2fy -\- c  =  0  (1) 

are  ax  -\-  hy  -]-  g  =  0    and    hx  -\-  by  ^f=0',^ 

whence  ,_^>^-//^         y^-^tzl^.  ro^ 

wnence  ^-j,^_^^        V  -  h'_ab'  ^^^ 

and  when  the  origin  is  moved  to  the  centre  without  changing  the 
direction  of  the  axes,  the  equation  (1)  reduces  to 

aai^  +  2hxy  +  hy'  +  c'  =  0,  (3) 

where  c'  =  gx!  -\-  jy'  +  c. 

Hence,  if  l^  —  ab  i^  0,  the  coordinates  of  the  centre  are  both 
finite,  and  this  transformation  is  possible. 

In  order  to  reduce  (3)  to  any  one  of  the  standard  forms 
(§§  119-121)  we  must  remove  the  term  Ihxy,  For  this  purpose 
we  turn  the  axes  through  a  certain  angle  B^  keeping  the  origin 
fixed. 

To  turn  the  axes  through  an  angle  0  we  substitute  for  x  and  i/, 
respectively  [§66,  (11)], 

X  cos  0  —  y  sin  0    and     x  sin  0  -\-  y  cos  0, 

Substituting  these  values  in  (3),  expanding  and  collecting 
terms,  we  have 

(a  cos^  ^  +  2A  sin  ^  cos  <?  +  6  sin^  d)x' 

+'  2  [(6  —  a)  sin  ^  cos  <?  +  /i(cos'  6  —  sin''  6)]  xy 

-\-(iasm^e  —  2hsin0co80-\-bcos'd)y''-\-c'=O.     (4) 

The  coefficient  of  xy  in  equation  (4)  will  vanish  if  0  be  so 
chosen  that 

2(6  —  a)  sin  ^  cos  <9  +  2;i(cos^  0  —  sin^  0)  =  0. 

*It  should  be  noticed  here  that  the  line  hx  +  by  +/=0  bisects  all  chords  parallel  to  the 
j/-axis.  (See  (4) ,  §  164.)  By  solving  equation  (1)  for  x,  It  can  also  be  shown  that  the  line 
ax  +  hy  +  g  =  0  bisects  all  chords  parallel  to  the  a;-axis.    (Qf.  also  §  84.) 


(9) 


167.]      GENERAL  EQUATION  OF  THE  SECOND  DEGREE.       289 

This  equation  is  equivalent  to 

(a  — 6)  sin 2^  =  2A  cos 2^.  (5) 

.-.     tan  2^  =  -^.*  (6) 

a  —  0 

Whence  sin  20  =  ±  -—=======-,  (7) 

|/(a  — 6)^+4/1^ 

and  co82^=±     .       ^~     =^.  (8) 

l/(^a—by-^4:h' 

Using  this  value  of  0  equation  (4)  takes  the  form 

a'x'-{-b'y'-\-c'=0,  ^ 

a'  h' 

where  a'=a  cos^  0  -\-2h  sin  ^  cos  ^  +  6  sin^  <?,  (10) 

and  h'  =  a  sin^  d —2h  sin  ^  cos  ^  +  6  cos=^ 0,  (11) 

Equation  (9)  is  therefore  the  required  result. 
The  values  of  a'  and  h'  may  be  expressed  in  terms  of  a,  6,  and  h 
as  follows: 

From  (10)  and  (11),  by  addition  and  subtraction,  we  obtain 

a'-^h'^a  +  h,  (12) 

and  a'— fe'  =  (a  — 6)cos2^ +2;isin2<?.  (13) 

Substituting  (7)  and  (8)  in  (13)  gives 

or. 

a'-b'^±Via-by+^^  =  ^^^.  (U) 

Whence,  from  (12)  and  (14), 

a'=:  j}a  +  6±  \/{a  —  hy  +  4:h?\,  ^        (15) 


and  h'  =  l]a-{-h^V{a  —  hy-^4:h^\,  (16) 

The  ambiguity  in  the  values  of  a'  and  h'  given  by  (15)  and  (16) 
may  be  removed  by  (14).  From  the  many  values  of  ^  which 
satisfy  (6)  we  will  agree  always  to  choose  that  one  which  lies  be- 
tween 0°  and  180°.     Then  0  will  always  be  an  aeute  angle,  and 

•Cy.  equations  (17) ,  §  109. 

20 


290 


GENERAL  EQUATION  OF  THE  SECOND  DEGREE. 


[167. 


sin  2  d  will  always  be  positive.     Therefore  it  follows  from  (14)  that 
a' —  h'  will  always  have  the  same  sign  as  h. 

It  is  also  worthy  of  notice  that  the  values  of  a'  and  6'  given  by 
(15)  and  (16)  are  the  two  roots  of  the  equation 

x'—(a-\-b)x  —  ih'  —  ab)=0.  (17) 

Hence  a'  and  b'  will  have  the  same  sign  or  opposite  signs  accord- 
ing as  A,^  —  a6  <  or  >  0 ;  i.  e.  according  as  the  curve  is  an  ellipse 
or  a  hyperbola  (§110). 

If  h"^ —  a6  =  0,  i.  e.  if  the  curve  is  a  parabola,  the  roots  of  (17) 
are  0  and  a  -f  6. 

Ex.    Transform  the  equation 

8x2  _|_  ^y.y  j^  5t,2_|_  ga.  _  162/  _  16  =  0 
to  the  staTidard  form^  and  conetruct  the  conic. 


y//V                              Y' 

X 
^ 

Y 

\        ^'''^ 

\ 

O         /         X 

\ 

— 

K 

The  equations  for  finding  the  centre  are 

4x4- 2/ +  2=0    and    2x  +  5y  =  S, 


Then 


gx^+fy'+c  =  -3Q. 


Therefore  the  equation  referred  to  parallel  axes  O^X^,  O^Y^  through  the 
centre  is 


Also       a^  =  i|a  +  5d=l/(a-6)2  +  4/i2J  =  ^(13dt5)=9or4, 
and  6^=^ja  +  6=Fi/(a— 5)2  +  4/iM=i(13=F5)=4or9. 


168.]      GENERAL  EQUATION  OF  THE  SECOND  DEGREE.       291 

Since  h  is  positive,  we  take  a''  =  9  and  h^=^. 

Hence  the  equation  of  the  curve  referred  to  its  own  axes  O^X^^,  O'Y"  as 
axes  of  coordinates  is 


4  +  9-^- 

Also, 

tan2^=     2/. 
a  —  5 

4 
3- 

Therefore  the  line  O'X"  must  be  drawn  so 

that 

Z^^O^^^^=itan 

-4. 

168.     When  W— ah  ^^. 

In  this  case  the  coordinates  of  the  centre  [(2),  §  167]  are  both 
infinite,  and  therefore  the  first  degree  terms  can  not  be  removed 
by  changing  to  a  new  system  of  axes  parallel  to  the  old. 

Since  the  second  degree  terms  now  form  a  perfect  square,  the 
general  equation  may  be  written 

(%  +  «^y+2^a:  +  2/i/  +  c  =  0,  (1) 

where  a  =  ^a,  /?  =  y^6,  a  has  the  same  sign  as  ^,  and  /?  is  always 

positive. 

/.    h^a^.  (2) 

First  Method,     From  equation  (6),  §  167,  we  have 

.      ^.         2/i  2a/S  2tan<? 

tan  2^  = -  =  -3— ^2  =  - —-2-.    '  (3) 

B  a 

.',    tan^  =  -,     or    — -.  (4) 

a  p  ^    ^ 

K  we  turn  the  axes  through  an  angle  given  by  either  of  these 
values  of  tan  ^,  the  coefficient  of  xy  in  the  new  equation  will 

vanish.     If  we  take  ^  =  tan~^l  —  —  J,  the  equation  of   the  new 

ic-axis  will  be 

ax-^^y  =  0.  (5) 

We  will  use  this  value,  and  will  agree  always  to  take  the  posi- 
tive direction  of  the  new  a:-axis  so  that  0  shall  be  numerically  less 
than  90°.  Then  0  will  be  positive  or  negative  according  as 
h  (or  a)  is  negative  or  positive,  and  we  have  from  (4) 

sin  0  =  ,         cos  0  =  — . 


292        GENERAL  EQUATION  OF  THE  SECOND  DEGREE.     [168. 

Hence,  to  turn  the  axes  through  an  angle  6  thus  chosen,  we 
must  substitute  for  x  and  y^  respectively  [§  66,  (11)], 


/^^  +  «2/      and    ~  "^  +  ^^ 
Substituting  the  expressions  (6)  in  (1)  gives 

?2^,,2,0      ^9  —  ^S^     LO      "S'  +  Z?/ 


(6) 


(a^  +  /52)  ?/2  +  2  -^  a;  +  2     "^    '  "  y  +  c  =  0.        (7) 

Completing  the  square  in  the  terms  containing  ?/,  equation 
(7)  may  be  reduced  to  the  form 

where  g,  c(.^  + /^-)-- («.  + /^/)- 

and        '  iTr-  «^  + '^^ 


If  now  the  origin  be  moved  to  the  point  (jff,  K),  equation 
(8)  will  take  the  standard  form 

y^=2-d^M=rX,  (9) 

Therefore  equation  (1)  represents  a  parabola  whose  axis  is 
parallel  to  the  line  (5),  and  whose  latus  rectum  is 

.     2(a/— /9gr) 

Second  Method.     Equation  (1)  may  be  written 

^ax  +  /52/  +  /l)^=  2(aA  — 5r)a^  +  2{^r—S)y  +  A»—  c,       (10) 

where  A  is  any  constant,  for  which  a  particular  value  will  now  be 
determined. 

We  observe  that  the  line  whose  equation  is 

ax-\-i3y^X  =  0  (11) 

is  parallel  to  the  axis  of  the  parabola  [see  (5)  above]  for  all 
values  of  A.     Hence  we  will  choose  A  so  that  the  straight  line 


168.]     GENERAL  EQUATION  OF  THE  SECOND  DEGREE.       293 

2{ak-g)x  +  K^^—S)y  ^l'—C  =  0  (12) 

shall  be  perpendicular  to  the  line  (11). 
The  Unes  (11)  and  (12)  will  be  at  right* angles  (§  48)  if 

a(a>l  — sr)-hW>l-/)  =  0, 

^  =  ^.  (13) 

With  this  value  of  A  equation  (10)  may  be  written 

(^ax+?y+iy=2"t^,i?x-ay  +  K),  (14) 


(16) 


Changing  the  linear  expressions  in  (14)  to  the  distance  form 
gives 

If  now  we  take  the  lines 

ax  +  ^y-{-k  =  0  (17) 

and  ^x  —  ay-\-K=0  (18) 

for  new  axes  of  x  and  y,  respectively,  the  new  equation  will  be 

(§  70  and  §  71) 

y^=2-£^M=x,  (19) 

Therefore,  if  we  assign  to  A  and  ^  the  values  given  by  (13)  and 
(15),  then  (17)  will  represent  the  axis  of  the  parabola  and  (18) 
the  tangent  at  the  vertex.  For  any  other  value  of  A,  (17)  is  a 
diameter  aud  (18)  is  the  tangent  at  its  extremity;  the  equation 
of  the  curve  (19)  will  then  be  expressed  in  terms  of  the  perpen- 
diculars upon  the  oblique  axes.  (See  Ex.  3,  §  71.)  In  all  cases 
the  curve  will  lie  on  the  positive  or  negative  side  of  the  line  (18) 
according  as  (a/ —  ^g)  is  positive  or  negative.^ 

*  In  the  investigations  of  §§  162-166,  and  again  in  §§  167, 168,  we  have  really  shown  by- 
independent  methods  that  the  general  equation  of  the  second  degree  always  represents  a 
conic  section.  In  the  latter  the  conditions  for  the  limiting  cases  have  not  been  pointed 
out.  The  student  should  do  this,  and  compare  the  results  of  both  these  discussions  with 
the  table  given  in  §  110. 


294 


GENERAL  EQUATION  OF  THE  SECOND  DEGREE. 


[168. 


Ex.    Find  the  standard  form  of  the  equation 

(43/  —  Sxy  —  20x  +  nOy  —  75  =  0. 


(1) 


"X 

Y 

\ 

1  "> 

\ 

x" 

^, 

7<- 

X 

2 

^ 

/^ 

\ 

\ 

/ 

V 

\ 

First  Method.    Take  iy  —  3x  =  0  as  the  new  x-axis  ;  i.  e.  turn  the  axes 
through  an  angle  6,  such  that  tan  ^  =  f ,  and  therefore  sin  ^  =  |,  cos  0  =  ^. 
Then  the  formulas  of  transformation  are 

ix'  —  3y' 


and 


x=x^  cos  6  —  y^  sin 


=  x^  sin  0-\-y^  cos  6  = 


5 

Sx'  + 


Substituting  these  values  in  equation  (1),  it  becomes 
2/^2-1-2x^+42/^—3  =  0, 


(2) 


or  (2/^+2)2  =  -2(a:'-|), 

which  is  the  equation  of  the  curve  referred  to  the  new  axes  OX^,  0Y\ 

Moving  the  origin  to  the  point  0^(|,  —  2),  with  respect  to  the  new  axes, 
we  obtain  from  (2)  the  required  equation 

y^^^=  —  2x''. 
Hence  the  curve  is  on  the  negative  side  of  the  y-axis  O'Y" . 
Second  Method.    The  given  equation  (1)  may  be  written 

(42/  -  3x  +  ?  )2  =  (20  —  6A)x  +  (8X  — 110)2/  +  ^'  +  75. 
"We  ¥rill  now  determine  a  so  that  the  two  lines 
Ay  —  Sx  +  ^  =  0 
and  (20  — 6A)x+(8A— 110)2/  +  ^^  +  75  =  0 

shall  be  at  right  angles. 


(3) 


(4) 

(5) 
(6) 


168.]     GENERAL  EQUATION  OF  THE  SECOND  DEGBBE.       295 

The  required  value  of  A  is  given  by  the  equation  (§  48) 
—  3(20  — 6A)  +  4(8A  — 110)=  0. 
.-.2  =  10. 
"With  this  value  of  "k  equation  (4)  becomes 

(42/  -  3x  + 10)=^  =  -  10(4x  +  32/  - 17^, 
^^  ^  ^42/-3^x  +  10y^_^^4.  +  3|-m^^  ^^^ 

Draw  the  lines 

42/  —  3x  +  10  =  0,  O'X^',  (8) 

and  4x  +  32/  — 17 J  =  0,  O'  Y",  (9) 

These  lines  are  at  right  angles.    If  we  take  (8)  as  the  new  x-axis  and 

<9)  as  the  new  2/-axis,  the  equation  of  the  curve  will  be  (§  70) 

2/^=-2x.  (10) 

Therefore  the  locus  is  a  parabola  whose  latus  rectum  is  2,  and  lies  on  the 

negative  side  of  the  line  (9). 

EXAMPLES. 

Construct  the  following  conies  by  transforming  the  equations  to  their 
standard  forms: 

1.  (42/  — 3x)2H-4(4x  +  32/)  =  0. 

2.  3x2 +  2x2/ +  32/^  =  8.  ^ 

3.  x2  —  6x2/ +  2/' =  16. 

4.  (3x  — 42/  — 12)2=15(4x  +  32/). 

5.  4x2— 24x2/ +  ll2/'^  —  16x  — 22/  — 89  =  0. 

6.  5x2  — 4x2/ +  82/2  — 24x  + 162/  — 4  =  0. 

7.  9x2— 12x2/  +  42/2  =  10(2x  +  32/  +  5). 

8.  3x2  — 2X2/  +  22/'— 16x  — 82/  +  8  =  0.    (See  Ex.,  §  163.) 

9.  2x2  +  4x^  +  52/2=36. 

10.  6x2+24x2/  — 2/^  +  502/  — 55  =  0. 

11.  x2  — 2x2/  +  2/'  — 5x  — 2/  — 2  =  0. 

12.  8x2  — 5x2/  — 42/2=34. 

13.  x2  — 6X2/  +  92/2  — 2x  +  62/  +  l  =  0. 

14.  4x2  +  4x1/ +  2/2  +  4X  — 32/ +  4=0. 

15.  2x2  +  x2/  +  32/2=23. 

16.  24X2/  +  72/'  — 6(6x  — lOt/- 9)  =  0. 


296  GENERAL   EQUATION   OF   THE   SECOND   DEGREE.  [169. 

17.  26x-'  —  20xy  +  Ay'-{-5x  —  2y  —  Q  =  0. 

18.  x^~2xy  —  y''  =  20. 

19.  {5y  +  12xy  =  102x. 

20.  2x''-^xy  —  6y^  —  bx  +  ny  —  S  =  0. 

21.  x''-^2xy  +  y^—i2x-\-2y  —  S=0. 

22.  xy  +  3x  —  5y  +  5  =  0, 

23.  2x-'-\-lxy-iy'-\-ix-j-1y  —  18\  =  0.    (See  Ex.,  §  165.) 

1 69.     The  standard  forms  of  the  equations  of  the  conic  sec- 
tions are 

y^=  4:axj 

2     I    "Ts"  ~  •^> 


a 


"'         f  _ 


2   =1, 


and  xy  =  K. 

If  the  axes  be  changed  in  any  manner  whatever,  the  new  equa- 
tions are  obtained  by  substituting  for  x  and  y  expressions  of  the 
form  (§  66) 

lx-\-my  -\-n,     and     Vx  +  m'y  -\-  n'. 

The  new  equations  ma^y  therefore  be  written 

{Vx  +  m'y  +  n'y —  4:a(lx  -f  m?/  +  n)  =:  0, 

h\lx  +  mi/  +  nf^  a\Vx  +  m'y  +  ny  —  a'h''  =  0, 

b\lx  +  w?/  -f  nf—aXl'x  H-  m'?/  +  ^')'  —  a'^'  =  0, 

and  {Ix  -\-my  -\-n)  (Vx  -f  m'?/  -\-n')  —  K—0. 

Hence,  if  the  left  member  of  an  equation,  the  right  member 
being  zero,  is  the  square  of  one  real  linear  expression  plus  some 
multiple  of  the  first  power  of  another  real  linear  expression,  the 
equation  represents  a  parabola ;  if  the  left  member  is  the  sum  of 
any  multiples  of  the  squares  of  two  real  linear  expressions  plus  a 
constant,  the  equation  represents  an  ellipse ;  if  the  left  member 
is  the  difference  of  some  multiples  of  the  squares,  or  (what  is  the 
same  thing)  the  product,  of  two  real  linear  expressions  plus  a 
constant,  the  equation  represents  a  hyperbola. 


169.]  GENERAL   EQUATION   OF   THE   SECOND    DEGREE.  297 

EXAMPLES. 

Transform  the  following  equations,  taking  as  new  axes  the  lines  repre- 
sented by  the  linear  expressions  which  the  equations  contain.  (See  §§ 
70, 71.)    Construct  the  curves: 

1.  (x  +  2)^  +  3(y-4)=0. 

2.  9(x- 3)-^ +  4(2/ +  5)2  =  36. 

3.  9(a;  +  5)»  — 16(2/  — 4)2  =  144. 

4.  (a:  +  3)(2/-5)  +  9  =  0. 

5.  (3x  — 4^  +  6)2  =  10(4x  +  32/  — 5). 

-  6.  4(3x  — 42/  — 8)2  + (4a; +  32/  — 3)^  =  20. 

7.  3(2/  +  2a;  — 2)2- 2(a;- 22/  — 6)2  =  30. 

8.  (12X  +  52/  — 9)2  +  52(5x  — 122/)  =  0. 

9.  3(2x  +  32/  +  6)2  +  4(3x  — 22/  +  6)2  =  i56.  ^ 

10.  (23/  —  4a;  +  3)2  —  4(x  +  22/  +  9)2  =  20. 

11.  (7x  — 242/)2  =  40(24a;  +  72/  — 21). 

12.  (a;  —  32/  + 12)2  _|_  (3^.  _^  j,  ^  2)2  =  40. 

13.  (2/  — ■/3a;  +  l)2  +  (2/+i/3a;  +  5)2=36. 

14.  (2/  — 2x  +  4)2  +  (32/  +  4x  — 3)2="100. 

15.  2(a;  +  2/  +  l)'  — 4(a;-^2/  +  4)2  =  32. 

16.  (x-22/  +  4)2  +  5(3x-2/  +  6)=0. 

17.  5(a;  +  22/  — 8)2  — 4(42/  — 3x)2  =  100, 

18.  (2/  +  2a;  +  3)2  +  2(x  +  32/  — 6)2  =  40. 

19.  (5x  +  12y  — 36)(12x  — 52/  — 15)  =  676. 

20.  (2/+/3x  +  3)2  — 12(2/— i/3x-2)=0. 

21.  2(2/-3a;  +  9)2  — 9(2x  +  ^  +  l)2  =  180. 

22.  (x  +  42/— 10)(4x  — 2/  — 4)  +  34  =  0. 

23.  (x+v/32/+i/3)(x— i/32/  +  2i/3)  +  16  =  0. 

24.  2(x  +  22/  +  6)2+(x  — 32/  — 6)2  =  40. 

25.  (x-32/  +  6)2  — 2(72/  — 24x  +  28)  =  0. 

26.  (X  — 22/  — 4)(x  +  32/  +  9)+20p/2  =  0. 

27.  (2x  +  32/  +  6)2  +  (4x-2/-4)2=221. 

28.  (3x  —  2/  +  9)'  —  2(2x  +  2/  — 2)2  =  90. 


298       GENERAL  EQUATION  OF  THE  SECOND  DEGREE.      [170. 

1 70.      The  equation  of  a  conic  through  given  points. 

The  general  equation  of  a  conic  contains  five  independent  con- 
stants (§  108).     When  the  equation  is  written 

ax'  +  2hxy  +  bf  +  2gx  +  2/^  +  c  =  0,  (1) 

these  constants  are  the  five  independent  ratios  between  the  six 
coefficients  a,  b,  c,  f,  g,  h.  It  follows,  therefore,  that,  in  general, 
one  and  only  one  conic  can  be  made  to  pass  through  any  five  given  points 
in  the  plane.  For,  if  we  substitute  for  x  and  y  in  (1)  the  coordi- 
nates of  the  five  given  points,  we  shall  have  five  linear  equations 
from  which  we  may  determine  uniquely  the  values  of  the  five 
ratios.  These  values  substituted  in  (1)  will  give  the  required 
equation. 

A  more  convenient  method  for  finding  the  equation  of  a  conic 
through  five  given  points  is  as  follows : 

Take  any  four  of  the  given  points 
and  connect  them  so  as  to  form  the 
quadrilateral  P1P2P3P4. 

Let  Wi,  W3,  W3,  Ui  be  the  equations 
of  the  lines  P,P,,  P,P„  P,P„  P,P,y 
respectively. 
Then 

u^u^  =  0     and     U2U^  =  0 

^p^  \      are   the   equations    of    two   conies 

whose  common  points  are  Pj,  P2,  P3,  Pi- 

.*.     U1U3  -{-  Xu^u^  —  0,  (3) 

whatever  the  value  of  X  may  be,  is  the  equation  of  a  conic  through 
the  four  points  P^  Pa,  P3,  P4. 

Equation  (3)  involves  one  arbitrary  parameter  X.  Its  locus 
can  therefore  be  made  to  satisfy  a  single  condition ;  e.  g.,  it  can 
be  made  to  pass  through  any  other  point  in  the  the  plane.  If  we 
substitute  for  x  and  y  in  (3)  the  coordinates  of  P5,  we  will  get  an 
equation  of  the  first  degree  in  I.  The  one  value  of  A  thus  deter_ 
mined  substituted  in  (3)  will  give  the  required  equation. 

Since  equation  (3)  contains  one  arbitrary  parameter,  it  follows 
that  an  infinite  number  oj  conies  can  be  made  to  pass  thro^igh  any  Jour 
given  points. 


171.]     GENERAL  EQUATION  OF  THE  SECOND  DEGREE.       299 

Four  points,  however,  are  suflBcient  to  determine  a  parabola. 
This  follows  from  the  fact  that  when  equation  (1)  represents  a 
parabola,  one  condition  connecting  the  coefficients  is  alwaj-s 
given,  viz.  /i^  —  ab  =  0. 

In  order  to  find  the  equation  of  a  parabola  determined  by  four 
given  points,  we  may  form  equation  (3)  as  before.  From  the 
coefficients  of  the  equation  thus  found  we  then  form  the  equation 

h'- — ab  =  0,  (4) 

Substituting  in  (3)  the  value  of  X  given  by  (4)  will  give  the 
equation  of  the  parabola  required.  Furthermore,  (4)  will  be 
an  equation  of  the  second  degree  in  X.  Therefore  two  parabolas  can 
be  made  to  pass  through  any  four  given  points  in  the  plane. 

Let  the  student  discuss  the  cases  when  three  or  more  of  the 
given  points  lie  on  the  same  straight  line. 

171.  To  find  the  equation  of  a  conic  having  two  given  lines  for  its 
asymptotes. 

Let  the  equations  of  the  asymptotes  be 

Ix  -\-my  -^n  =  0     and     Vx  -\-  m'y  -f-  n'  =  0. 

Then,  since  the  equation  of  the  conic  differs  from  the  equation 
of  its  asymptotes  only  by  a  constant  (§  117  and  §  146),  the  re- 
quired equation  is 

{Ix  -\-my^n)  (I'x  +  m'i/  +  n')  +  A  ==  0,  (1 ) 

where  A  may  have  any  value  whatever. 

The  conic,  therefore,  is  not  uniquely  deteimined,  but  can  still 
be  made  to  satisfy  one  more  condition ;  that  is,  it  can  be  made  to 
pass  through  a  given  point,  or  touch  a  given  line,  etc. 

Since  equation  (1)  involves  only  on^  arbitrary  parameter,  hav- 
ing given  the  asymptotes  of  a  conic  is  equivalent  to  having  given 
four  of  the  conditions  which  a  conic  can  be  made  to  satisfy. 

How  many  conditions  can  a  conic  be  made  to  satisfy, 

(1)  if  the  centre  is  given? 

(2)  if  the  two  foci  are  given? 

(3)  if  one  focus  and  the  corresponding  directrix  are  given? 

(4)  if  the  position  of  the  axes  is  given? 

(5)  if  the  two  directrices  are  given? 


300  GENERAL   EQUATION   OF   THE   SECOND   DEGREE.  [171. 

Examples  on  Chapter  XII. 

Find  the  equation  of  the  conies  through  the  points 

I.  (4,  3),  (2,  5),  (-2,  3),  (0,  -  2),  (2,  - 1). 
Z    (4,  3),  (0,  0),  (-2,  3),  (0,-2),  (2,  -1). 

3.  (a,  a),  (±a,  0),  (0,  iha). 

4.  (1,  2),  (-2,  4),  (-3,  -1),  (1,  ^2),  (2,-1). 

5.  (-1,-1),  (2,  3),  (-3,  5),  (-3,  -  2),  (1,  -7). 
Find  the  equations  of  the  parabolas  through  the  points 

6.  (4,  6),  (2, -4),  (-2,0),  (-3,  6). 

7.  (1,5),  (4,2),  (-3,-1),  (1,-1). 

8.  (2,0),  (-4,-2),  (2,  7),  (0,-3). 

9.  Find  the  equation  of  a  conic  through  the  origin  and  having  for  its 
asymptotes  the  lines 

X  — 22/  +  4  =  0    and    Sx-\-y—2  =  0. 

10.  Find  the  equation  of  a  conic  through  the  point  (1,  3),  if  the  equa- 
tions of  its  asymptotes  are 

2x  —  3y  —  l  =  0    and    6y+3x  —  S  =  0. 

II.  What  is  the  equation  of  a  conic  touching  the  x-axis,  if  its  asymp- 
totes are  the  lines 

2x  +  2/— 2  =  0    and    x  — 32/4-5  =  0? 

12.  What  is  the  equation  of  the  conic  touching  the  line  y-\-2x-\-i  =0, 
and  having  for  asymptotes 

X  — 2/  — 1  =  0    and    x  +  3y  —  6  =  0? 

13.  Find  the  equation  of  the  conic  wMch  passes  through  the  point 
(2,  —  2)  and  has  the  same  asymptotes  as 

5x2  _  2xy  —  32/2  +  4x  +  122/  =  0. 

14.  Find  the  asymptotes  of  the  hyperbola 

3x2  — 7x2/  — 62/2  — 8x  +  22/  — 4  =  0; 
find  also  the  equation  of  the  conjugate  hyperbola. 

15.  The  equation  of  a  given  hyperbola  is 

3x2  _  ^^y  _Sy2_i2x—y--2  =  0. 
Find  the  equation  of  its  conjugate. 


171.]  GEINERAL   EQUATION   OF   THE   SECOND   DEGREE.  301 

16.  Show  that  the  product  of  the  semi-axes  of  the  conic 

ax^-\-2hxy  +  by^-\-2gx-\-2fy-\-c  =  0 
—  C  —  c^ —A 

where  A  is  the  discriminant,  a',  6'  are  given  by  (15)  and  (16),  §  167,  and 
c'  by  (9),  §  109. 

5 

17.  Show  that     , — ^  is  the  product  of  the  semi-axes  of  the  conic 

a;2_2a;2/  — 32/2  — 2x  +  10^/  — 8  =  0. 
Show  also  that  the  equation  of  the  axes  of  this  conic  is 
X  — y'^r\-xy  —  5a;-f-5  =  0. 

18.  Show  that  the  squares  of  the  semi-axes  of  the  conic 

ax^-\-2hxy  +  by' -\-2gx  +  2fy  +  c  =  0 
are  2  /\-^{ab  —  h'')  \a-\-b ±  V (a  —  by -\-  iJi'  \ , 

where  A  is  the  discriminant. 

19.  Show  that  the  lengths  of  the  semi-axes  of  the  conic 

ax'^-{-2hxy-\-ay''  =  K, 


Th  """  \^^h' 


are       ^  -\l — ^^    and 

\  a 

respectively,  and  that  their  equation  is  x'^  —  2/^  =  0. 

20.  Show  that  of  all  the  conies  which  pass  through  the  points  of  inter- 
section of  two  conies  only  two  are  parabolas. 

21.  Find  the  equations  of  the  two  parabolas  which  pass  through  the 
common  points  of 

x'  —  y'  =  l    and    x''-\-y'  —  2x  =  4. 

22.  Show  that  all  conies  passing  through  the  intersections  of  two  rect- 
angular hyperbolas  are  rectangular  hyperbolas. 

23.  If  two  rectangular  hyperbolas  intersect  in  four  real  points,  the  line 
joining  any  two  of  the  points  of  intersection  is  perpendicular  to  the  line 
joining  the  other  two.     (Ex.  22.) 

24.  Show  that  if 

ax'  -f  2hxy  +  by'  —  1    and    a^x'  +  2h^xy  -f-  b^y"^  —  1 
represent  the  same  conic,  and  the  axes  are  rectangular,  then 
(a  —  by  +  4/i2  =  (a^—  b'Y  +  W. 

25.  Show  that  for  all  positions  of  the  axes,  so  long  as  they  remain  rect- 
angular and  the  origin  is  unchanged,  the  value  of  g'  -\-p  in  the  equation 
ax'  -f  2/1X1/  -f  W^ -\-2gx  -\-2fy  -\-c  =  0  is  constant. 


302       GENERAL  EQUATION  OF  THE  SECOND  DEGREE.      [171. 

26.  If  ax^  +  2hxy  -\-  by^  =  1  and  a^x"^  +  2h^xy  +  b^y^  —  1  are  the  equations 
of  two  conies,  then  will  aa'-\-  2hW-\-  bb'  be  unaltered  by  any  change  of 
rectangular  axes. 

27.  The  polars  of  a  point  P  with  respect  to  two  given  circles  meet  in  Q ; 
if  P  moves  along  a  given  straight  line,  show  that  the  locus  of  Q  is  a  hyper- 
bola whose  asymptotes  are  perpendicular  to  the  given  line  and  to  the  line 
joining  the  centres  of  the  circles. 

28.  A  variable  circle  always  passes  through  a  fixed  point  O  and  cuts  a 
conic  in  P,  Q,  i2,  fif;  show  that 

OP' oq,' OR' OS 

(radius  of  circle)''' 
is  constant. 

29.  If  OPQ,  and  OP'Qf  are  two  straight  lines  which  are  always  parallel 
to  two  fixed  straight  lines,  and  meet  a  given  conic  in  P,  Q  and  P^,  Q^,  re- 
spectively, then  will  the  ratio  >-)p/  .^q/  ^®  ^^®  same  for  all  positions  of  0. 
Find  the  value  of  this  ratio  by  putting  O  at  the  centre  of  the  conic. 

30.  Show  that  the  conic  jS  +  'kv?'  —  0  touches  the  conic  fif  =  0  where  the 
latter  is  cut  by  the  straight  line  u  —  ^. 

31.  If  two  conies  have  their  axes  parallel,  a  circle  will  pass  through  their 
points  of  intersection. 

32.  Two  conies  are  said  to  be  similar  and  similarly  placed  when  their 
axes  are  parallel  and  they  have  the  same  eccentricity.    (§  116. ) 

Hence  show  that  the  two  conies 

ax^  +  2/1X2/ +  62/2  +  2£rx  +  2/2/ +  c  =  0 
and  a'x"  +  IWxy  +  b'y^  +  Ig'x  +  2fy  +  c'=  0 

will  be  similar  if 

h^  —  ab      W—a'b'* 

Show  also  that  they  will  be  similarly  placed  if 

h      ^      W 
a  —  b~  a'—  b'' 

Then  show  that  they  will  be  both  similar  and  similarly  placed  if 

a  _h  _b 

[Use  equations  (15)  and  (17)  of  §  109.] 


3. 

tan  ^  cot  ^  =  1. 

4. 

^     a      sin  ^ 
cos  0 

5. 

sin2^  +  cos2^=:l. 

6. 

sec2<9  — tan2^  =  l. 

7. 

csc2(9  — cot2^  =  l. 

APPEKDIX. 

TRIGONOMETRICAL  FORMUL.2E. 

1.  sin  (9  esc  ^=1.  8.  sin(— <9)  =  —  ainO, 

2.  cos  ^  sec  ^  =  1.  9.  cos  (—  6)  =  cos  6, 

10.  sin(90°  ±^)  =  cos(9. 

11.  cos  (90°  ±  ^)  =  T  sin  (9. 

12.  8in(180°  ±<9)=  Tsin6/. 

13.  cos(180°±(9)  =  — cos^. 

14.  sin  (270°  ±0)  =  —  cos  6, 

15.  cos  (270°  ±6)=:  ±  sin  (?. 

16.  sin  (0  ±  6')  =  sin  6  cos  ^'  ±  cos  <?  sin  (9'. 

17.  cos  {0  ±  e^)  z=  cos  0  cos  ^^  T  sin  ^  sin  ^^. 

Ao     4.     /a  .  fl/N      tan<9±tan^'  .„     ^      _„       2tan(9 

18.  tan  (^  ±  ^^)  =  . — 7 — in^ — tt/.  19-    tan  2^=  ;i — ^. 

^  It  tan ^ tan ^^  1  — tan^^ 

on         4./0  .  fl/^     cot^cot^'Tl  ^  „,  .„.      eot2/9  — 1 

20.      cot  (/9  ±  ^^)  =  --r;r- ttt  •  21.      COt  26  =  —5 — — . 

^  ^       cot  tf^  ±  cot  f/  2  cot  ^ 

22.  sin  2^  =  2  sin  ^  cos  ^. 

23.  cos  20  =  cos^^  —  sin'^  =  2  cos^l?  — 1  =  1  —  2  sin«0. 

24.  sin  ^6  =  /^(l— cos^).  25.    cos  ^(9  =  1/^(1+ cos ^) 

26.  sin  ^  +  sin  <9/  =  2  sin  i(^  +  6^)  cos  K^  —  ^'). 

27.  sin0  — sin^^  =  2cos  J((9  +  ^/)sinK^  — ^'). 

28.  cos  d-\- cos  6^  =  2  cos  ^{6  +  ^/)  cos  i(<9  —  0'). 

29.  cos/9  — cos^^  =  — 2sinK^  +  ^0sinK^  — ^0- 

In  any  plane  triangle 

qo     ^^^^  —  ^^^^  —  ^^^^  ^\     ^Lt?  —  tan  }(A  -\-  B) 

a     ~     b     ~     <c    *  'a  —  b~  tan  i(^  —  B)' 

32.    a''=b^  +  c^  —  2bccoaA,  33.    Area  =  i6c  sin  ^. 


■1 


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